# Asymptotic total ergodicity for actions of $\mathbb{F}[t]$ and polynomial
configurations over finite fields and rings

Ethan Ackelsberg  School of Mathematics, Institute for Advanced Study,
Princeton, NJ 08540<EMAIL_ADDRESS>and  Vitaly Bergelson  Department of
Mathematics, Ohio State University, Columbus, OH 43210<EMAIL_ADDRESS>state.edu

###### Abstract.

We obtain new combinatorial results about polynomial configurations over
finite fields and rings by utilizing the phenomenon of _asymptotic total
ergodicity_ (previously studied for actions of $\mathbb{Z}$ on modular rings
$\mathbb{Z}/N\mathbb{Z}$ in [BB23]) in the context of actions of the
polynomial ring $\mathbb{F}[t]$ over a finite field $\mathbb{F}$. We prove
that a sequence of quotient rings $\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]$,
$Q_{n}(t)\in\mathbb{F}[t]$, is asymptotically totally ergodic if and only if
the minimal degree of the irreducible factors of $Q_{n}$ diverges to infinity
as $n\to\infty$. We then show that asymptotic total ergodicity of a sequence
of quotient rings $\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]$ leads to asymptotic
equidistribution of polynomial sequences in subgroups of
$\left(\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t],+\right)$. This has several
combinatorial consequences:

  1. (1)

We prove a power saving bound for the Furstenberg–Sárközy theorem over finite
fields of fixed characteristic: given an _intersective_ polynomial
$P(x)\in(\mathbb{F}_{p}[t])[x]$, there exists $\gamma=\gamma(P)>0$ such that
if $A\subseteq\mathbb{F}_{q}$, $q=p^{n}$, does not contain distinct $a,b\in A$
with $b-a=P(y)$ for some $y\in\mathbb{F}_{q}$, then $A\ll_{P}q^{1-\gamma}$.
This complements recent work of Li and Sauermann [LS22], where they obtain
power saving bounds under the assumption $P(0)=0$.

  2. (2)

Under a natural equidistribution condition on $P(x)\in(\mathbb{F}[t])[x]$, we
prove the following enhancement of the Furstenberg–Sárközy theorem in the
presence of asymptotic total ergodicity. Suppose the sequence of quotient
rings $\left(\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]\right)_{n\in\mathbb{N}}$ is
asymptotically totally ergodic. Then for any $\delta>0$ and any two sets
$A,B\subseteq\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]$ with
$|A||B|\geq\delta\left|\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]\right|^{2}$ and $n$
sufficiently large, there exist $x,y\in\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]$
with $x\in A$ and $x+P(y)\in B$. Moreover, the values
$x,y\in\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]$ for which $x\in A$ and $x+P(y)\in
B$ obey the “correct” statistical behavior as $n\to\infty$:

$\displaystyle\frac{\left|\left\\{(x,y)\in\left(\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]\right)^{2}:x\in
A,x+P(y)\in
B\right\\}\right|}{\left|\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]\right|^{2}}=\frac{|A||B|}{\left|\mathbb{F}[t]/Q_{n}(t)\mathbb{F}[t]\right|^{2}}+o_{n\to\infty}(1).$

We also show that, in the absence of asymptotic total ergodicity and an
equidistribution condition on the polynomial $P$, one cannot hope for such a
refinement of the Furstenberg–Sárközy theorem.

  3. (3)

We establish partition regularity of families of polynomial equations over
finite fields. For example, we are able to prove: if $P(x)\in\mathbb{Z}[x]$
with $P(0)=0$, then for any $r\in\mathbb{N}$, there exists $N=N(P,r)$ and
$c=c(P,r)>0$ such that if $q>N$ and $\mathbb{F}_{q}=\bigcup_{i=1}^{r}{C_{i}}$,
then there are at least $cq^{2}$ monochromatic solutions to the equation
$P(x)+P(y)=P(z)$.

Interactions between finitary mathematics (combinatorics in finite fields and
rings) and infinitary mathematics (ergodic theory, equidistribution, and Loeb
measure spaces) play a central role throughout the paper.

###### Key words and phrases:

Finite fields, Furstenberg–Sárközy theorem, total ergodicity,
equidistribution, partition regularity, Loeb measure

###### 2020 Mathematics Subject Classification:

11B30 (11T06, 37A25, 05D10)

###### Contents

  1. 1 Introduction
  2. 2 Asymptotic total ergodicity
  3. 3 Asymptotic projection theorem
  4. 4 Power saving bounds for the Furstenberg–Sárközy theorem in characteristic $p$
  5. 5 Proof of equivalences
  6. 6 Partition regularity of polynomial equations over finite fields

##  1\. Introduction

Our starting point is the following classical result:

######  Theorem 1.1 (Furstenberg–Sárközy theorem [F77, S78]).

Let $P(x)\in\mathbb{Z}[x]$ be a nonzero polynomial with $P(0)=0$. For any
$\delta\in(0,1)$, there exists $N=N(P,\delta)\in\mathbb{N}$ such that any
subset $A\subseteq\\{1,\dots,N\\}$ of size $|A|\geq\delta N$ contains distinct
elements $a,b\in A$ with $b-a=P(n)$ for some $n\in\mathbb{Z}$.

The assumption that $P(0)=0$ can weakened as follows. Call a polynomial $P$
_intersective_ if $P$ has a root mod $q$ for every $q\in\mathbb{N}$. It
follows from the work of Kamae and Mendès France [KM78, Example 3] that the
conclusion of Theorem 1.1 holds if and only if $P$ is intersective.

A version of the Furstenberg–Sárközy theorem holds also in a finite
characteristic setting. The following result is a consequence of [BL16,
Theorem 9.2] together with (a corrected version of) the remark111In the remark
following Theorem 9.5 in [BL16], intersective polynomials are defined as
polynomials $P(x)\in(\mathbb{F}[t])[x]$ such that for any finite index
subgroup $\Lambda\leq(\mathbb{F}[t],+)$, there exists $m\in\mathbb{F}[t]$ such
that $P(mn)\in\Lambda$ for every $n\in\Lambda$. The definition of intersective
given in item (i) of Theorem 1.2 is different and deals with a wider class of
polynomials but is the correct notion in order to get the desired “if and only
if” conclusion. An example of an intersective polynomial that does not fit the
condition in [BL16] is $P(x)=x+t$. There is no multiple $m\in\mathbb{F}[t]$
for which $P(mn)$ always belongs to the subgroup $t^{2}\mathbb{F}[t]$ of index
$|\mathbb{F}|^{2}$. However, $P(-t)=0$, so $P$ is intersective (according to
our definition). following Theorem 9.5 in [BL16]:

######  Theorem 1.2.

Let $\mathbb{F}$ be a finite field with $q$ elements. The following are
equivalent for a polynomial $P(x)\in(\mathbb{F}[t])[x]$:

  1. (i)

$P$ is _intersective_ : for any $Q(t)\in\mathbb{F}[t]\setminus\\{0\\}$, there
exists $n\in\mathbb{F}[t]$ such that $P(n)\equiv 0\pmod{Q}$;

  2. (ii)

for any $\delta\in(0,1)$, there exists $N=N(P,\delta)\in\mathbb{N}$ such that
any subset $A\subseteq\mathbb{F}[t]$ of elements of degree $<N$ with
$|A|\geq\delta q^{N}$ contains distinct $a,b\in A$ with $b-a=P(n)$ for some
$n\in\mathbb{F}[t]$.

A consequence of Theorem 1.2 is a Furstenberg–Sárközy type theorem over large
finite fields:

######  Corollary 1.3.

Let $p$ be a prime number, and let $P(x)\in(\mathbb{F}_{p}[t])[x]$ be an
intersective polynomial. Then for any $\delta\in(0,1)$, there exists
$N=N(P,\delta)\in\mathbb{N}$ such that if $k\geq N$ and
$A\subseteq\mathbb{F}_{p^{k}}$ with $|A|\geq\delta p^{k}$, then $A$ contains
distinct $a,b\in A$ with $b-a=P(n)$ for some $n\in\mathbb{F}_{p^{k}}$.

Recent work of Li and Sauermann [LS22], building on earlier work of Green
[G17] using the Croot–Lev–Pach [CLP17] polynomial method, establishes
quantitative improvements of Theorem 1.2 and Corollary 1.3 under the
assumption $P(0)=0$.

######  Theorem 1.4 ([LS22], Theorem 1.4 and Corollary 1.5).

For $q,d\in\mathbb{N}$, let

$\displaystyle
t_{q,d}=\inf_{0<x<1}{\frac{1+x+\dots+x^{q-1}}{x^{\frac{1}{2}(q-1)(1-1/(dd^{\prime}))}}},$

where

$\displaystyle d^{\prime}=\min\\{d,(q-1)(1+\log_{q}{d})\\}.$

  1. (i)

Fix $q$ and a polynomial $P(x)\in\mathbb{F}_{q}[x]$ of degree $d$ with
$P(0)=0$. If $A\subseteq\mathbb{F}_{q}[t]$ is a set of polynomials of degree
less than $N$ and $A$ does not contain distinct $a,b\in A$ with $b-a=P(x)$ for
some $x\in\mathbb{F}_{q}[t]$, then222The notation $a(N)\ll b(N)$ means that
there is a constant $C>0$ such that $|a(N)|\leq C|b(N)|$ for all
$N\in\mathbb{N}$. Subscripts on $\ll$ denote on which parameters the constant
$C$ depends.

$\displaystyle|A|\ll_{q,d}t_{q,d}^{N}.$

  2. (ii)

Fix a prime $p$ and a polynomial $P(x)\in\mathbb{F}_{p}[x]$ of degree $d$ with
$P(0)=0$. If $A\subseteq\mathbb{F}_{p^{k}}$ does not contain distinct $a,b\in
A$ with $b-a=P(x)$ for some $x\in\mathbb{F}_{p^{k}}$, then

$\displaystyle|A|\ll_{p,d}t_{p,d}^{k}.$

In this paper, we produce power saving bounds for the Furstenberg–Sárközy
theorem over finite fields by a different method. The bounds we obtain are
different from those in Theorem 1.4. In some cases, our bounds are stronger,
while in other cases, ours are weaker; see Remark 1.13 below. Our approach
draws inspiration from infinitary sources. These are: equidistributional
results for polynomial sequences defined over $\mathbb{F}[t]$ and the
phenomenon of _asymptotic total ergodicity_ for actions of $\mathbb{F}[t]$. As
a consequence of our approach, our results apply in a more general setting
than finite fields of characteristic $p$, including quotient rings
$\mathbb{F}[t]/Q(t)\mathbb{F}[t]$ under some conditions on
$Q(t)\in\mathbb{F}[t]$. The phenomenon of asymptotic total ergodicity for
$\mathbb{Z}$-actions was previously explored in [BB23], where similar
Furstenberg–Sárközy-type results are proved in the setting of modular rings
$\mathbb{Z}/N\mathbb{Z}$ when all prime factors of $N$ are sufficiently large.
The results of this paper are natural analogues of the results in [BB23].
Where appropriate, we note the correspondences between our setting (dealing
with $\mathbb{F}[t]$-actions and quotient rings
$\mathbb{F}[t]/Q(t)\mathbb{F}[t]$) and the more familiar setting of
$\mathbb{Z}$-actions and modular rings $\mathbb{Z}/N\mathbb{Z}$. However, some
caution is needed, as our finite characteristic setting introduces new
complications. Namely, the distributional behavior of a polynomial whose
degree $d$ exceeds the characteristic $p$ is more sophisticated than the
behavior of polynomials over $\mathbb{Z}$ (see, e.g., Theorem 1.9 below), and
this creates additional difficulties in our analysis.

Before stating our results, we fix some notation. Let $\mathbb{F}$ be a finite
field of characteristic $p$. We denote the set of monic polynomials over
$\mathbb{F}$ by $\mathbb{F}[t]^{+}$. Every element $Q(t)\in\mathbb{F}[t]^{+}$
has a unique factorization (up to reordering) into monic irreducibles
$Q(t)=Q_{1}(t)^{s_{1}}\dots Q_{r}(t)^{s_{r}}$. We denote the quotient ring
$\mathbb{F}[t]/Q(t)\mathbb{F}[t]$ by $\mathbb{F}[t]_{Q}$, and we have an
isomorphism

(1.1)
$\displaystyle\mathbb{F}[t]_{Q}\cong\mathbb{F}[t]_{Q_{1}^{s_{1}}}\times\dots\times\mathbb{F}[t]_{Q_{r}^{s_{r}}}$

by the Chinese remainder theorem. Note that when $Q$ is irreducible,
$\mathbb{F}[t]_{Q}$ is a finite field of characteristic $p$. Moreover, any
finite field of characteristic $p$ can obtained as $\mathbb{F}_{p}[t]_{Q}$ for
some irreducible element $Q(t)\in\mathbb{F}_{p}[t]^{+}$. The decomposition of
the ring $\mathbb{F}[t]_{Q}$ for general $Q(t)\in\mathbb{F}[t]^{+}$ given in
(1.1) is parallel to the situation with modular rings, where the Chinese
remainder theorem gives an isomorphism

$\displaystyle\mathbb{Z}/N\mathbb{Z}\cong\mathbb{Z}/p_{1}^{s_{1}}\mathbb{Z}\times\dots\times\mathbb{Z}/p_{r}^{s_{r}}\mathbb{Z}$

for $N=p_{1}^{s_{1}}\dots p_{r}^{s_{r}}$.

We define an absolute value on $\mathbb{F}[t]$ by
$\left|c_{d}t^{d}+\dots+c_{1}t+c_{0}\right|=|\mathbb{F}|^{d}$ if $c_{d}\neq
0$. Equivalently, for any $Q(t)\in\mathbb{F}[t]$, $|Q|$ is the cardinality of
the quotient ring $\mathbb{F}[t]_{Q}$. For $Q=Q_{1}^{s_{1}}\dots
Q_{r}^{s_{r}}\in\mathbb{F}[t]^{+}$, we set $\text{lpf}(Q)=\min_{1\leq i\leq
r}{|Q_{i}|}$ to be the size of the least prime factor of $Q$.

For a finite set $S$ and a function $f:S\to\mathbb{C}$, we write

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
S}{f(x)}=\frac{1}{|S|}\sum_{x\in S}{f(x)}.$

For $r\geq 1$, we define the $L^{r}$-norm on $S$ by

$\displaystyle\left\|f\right\|_{L^{r}(S)}:=\left(\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
S}{\left|f(x)\right|^{r}}\right)^{1/r}.$

Recall that a measure-preserving $\mathbb{Z}$-system333Given an abelian group
$\Gamma$, a _measure-preserving $\Gamma$-system_ is a quadruple
$\left(X,\mathcal{X},\mu,(T_{g})_{g\in\Gamma}\right)$, where
$(X,\mathcal{X},\mu)$ is a probability space, and $(T_{g})_{g\in\Gamma}$ is an
action of $\Gamma$ on $(X,\mathcal{X},\mu)$ by measure-preserving
transformations. $(X,\mathcal{X},\mu,T)$ is _totally ergodic_ if for every
$m\in\mathbb{Z}\setminus\\{0\\}$, $T^{m}$ is ergodic. By analogy, we say that
a measure-preserving $\mathbb{F}[t]$-system
$\left(X,\mathcal{X},\mu,(T_{n})_{n\in\mathbb{F}[t]}\right)$ is _totally
ergodic_ if for every $m\in\mathbb{F}[t]\setminus\\{0\\}$, the action
$(T_{mn})_{n\in\mathbb{F}[t]}$ is ergodic. Our first result provides a
finitization of the phenomenon of total ergodicity. A similar result for
$\mathbb{Z}$-actions and the quotient rings $\mathbb{Z}/N\mathbb{Z}$ appears
in [BB23].

######  Theorem 1.5.

Let $(Q_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathbb{F}[t]^{+}$. The
following are equivalent:

  1. (i)

The sequence of quotient rings $\mathbb{F}[t]_{Q_{n}}$ is asymptotically
totally ergodic: for any $m\in\mathbb{F}[t]\setminus\\{0\\}$,

$\displaystyle\sup_{f_{n}:\mathbb{F}[t]_{Q_{n}}\to\mathbb{D}}{\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q_{n}}}{f_{n}(x+my)}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q_{n}}}{f_{n}(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q_{n}})}}\xrightarrow[n\to\infty]{}0.$

  2. (ii)

$\text{lpf}(Q_{n})\to\infty$.

We prove Theorem 1.5 in Section 2.

Using the spectral theorem for unitary actions of $\mathbb{F}[t]$ and
equidistributional results for polynomial sequences over $\mathbb{F}[t]$, one
may show the following:

######  Theorem 1.6.

Let $P(y)\in(\mathbb{F}[t])[y]$ be a polynomial with zero constant term. Then
for any totally ergodic system
$\left(X,\mathcal{X},\mu,(T_{n})_{n\in\mathbb{F}[t]}\right)$, any Følner
sequence444 A _Følner sequence_ in $\mathbb{F}_{p}[t]$ is a sequence
$(\Phi_{N})_{N\in\mathbb{N}}$ of finite subsets of $\mathbb{F}_{p}[t]$ such
that, for any $n\in\mathbb{F}_{p}[t]$,
$\displaystyle\lim_{N\to\infty}{\frac{\left|(\Phi_{N}+n)\triangle\Phi_{N}\right|}{|\Phi_{N}|}}=0.$
Examples include
$\Phi_{N}=\left\\{c_{N-1}t^{N-1}+\dots+c_{1}t+c_{0}:c_{i}\in\mathbb{F}_{p}\right\\}$
(the set of all polynomials over $\mathbb{F}_{p}$ of degree $<N$) and
$\Phi^{\prime}_{N}=\left\\{t^{N}+c_{N-1}t^{N-1}+\dots+c_{1}t+c_{0}:c_{i}\in\mathbb{F}_{p}\right\\}=\Phi_{N}+t^{N}$
(the set of all monic polynomials of degree $N$).
$(\Phi_{N})_{N\in\mathbb{N}}$, and any $f\in L^{2}(\mu)$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{T_{P(n)}f}}=\pi_{P}(f),$

where $\pi_{P}$ is the orthogonal projection onto the space

$\displaystyle\left\\{g\in L^{2}(\mu):T_{P(n)}g=g~{}\text{for
all}~{}n\in\mathbb{F}[t]\right\\}.$

######  Remark 1.7.

(1) A similar result with $\mathbb{F}[t]$ replaced by a countably infinite
field was obtained by Larick in [L98, Theorem 1.1.1].

(2) For $\mathbb{Z}$-actions, the corresponding version of Theorem 1.6 is
simpler. Namely, for any polynomial $P(y)\in\mathbb{Z}[y]$, any totally
ergodic $\mathbb{Z}$-system $(X,\mathcal{X},\mu,T)$, any Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$ in $\mathbb{Z}$, and any $f\in L^{2}(\mu)$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{T^{P(n)}f}}=\int_{X}{f~{}d\mu}.$

The presence of the projection $\pi_{P}$ in Theorem 1.6 rather than
$\int_{X}{f~{}d\mu}$ is a reflection of the more intricate distributional
behavior of polynomials over $\mathbb{F}[t]$. The situation where the
projection $\pi_{P}f$ is equal to $\int_{X}{f~{}d\mu}$ can be characterized by
an equidistributional assumption on the polynomial
$P(y)\in(\mathbb{F}[t])[y]$; see Proposition 1.14 below.

Our goal is to prove an asymptotic version of Theorem 1.6 and to deduce from
it new combinatorial results over finite fields (and rings of the form
$\mathbb{F}[t]_{Q}$). Before formulating our result, we sketch a proof of
Theorem 1.6, which will serve as a model for our finitary results.

By the spectral theorem for actions of $\mathbb{F}[t]$ by unitary operators on
a Hilbert space, we may work with the Hilbert space
$\mathcal{H}=L^{2}\left(\widehat{\mathbb{F}[t]},\sigma\right)$, where $\sigma$
is a positive Borel measure on the dual group $\widehat{\mathbb{F}[t]}$, and
the unitary action $(T_{n})_{n\in\mathbb{F}[t]}$ is represented by the
multiplication operators $(U_{n}h)(\chi)=\chi(n)h(\chi)$ for $h\in\mathcal{H}$
and $\chi\in\widehat{\mathbb{F}[t]}$.

Rather than working with $\widehat{\mathbb{F}[t]}$ as the abstract dual group
of $\mathbb{F}[t]$, it will be convenient to work with the dual group in a
more concrete form. Let $\mathbb{F}(t)$ be the field of rational functions
$\mathbb{F}(t)=\left\\{\frac{m}{n}:m,n\in\mathbb{F}[t],n\neq 0\right\\}$.
Extending the absolute value we defined on $\mathbb{F}[t]$ to $\mathbb{F}(t)$
by $\left|\frac{m}{n}\right|=\frac{|m|}{|n|}$, the completion of
$\mathbb{F}(t)$ is the field
$\mathbb{F}((t^{-1}))=\left\\{\sum_{j=-\infty}^{N}{c_{j}t^{j}}:N\in\mathbb{Z},c_{j}\in\mathbb{F}\right\\}$.
We think of $\mathbb{F}(t)$ and $\mathbb{F}((t^{-1}))$ as natural analogues of
the rational numbers $\mathbb{Q}$ and the real numbers $\mathbb{R}$,
respectively.

In the characteristic zero setting, the dual group of the integers is
isomorphic to the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. A similar result
is true in our setting: the dual group $\widehat{\mathbb{F}[t]}$ is isomorphic
to $\mathbb{F}((t^{-1}))/\mathbb{F}[t]$. In particular, every character
$\chi:\mathbb{F}[t]\to\mathbb{C}$ takes the form

$\displaystyle\chi(n)=e(nx)$

for some $x\in\mathbb{F}((t^{-1}))/\mathbb{F}[t]$, where

$\displaystyle
e\left(\sum_{j=-\infty}^{N}{c_{j}t^{j}}\right)=\chi_{0}(c_{-1})$

and $\chi_{0}$ is a fixed nontrivial character on $\mathbb{F}$.

A word of caution: with the objects discussed above, we have an isomorphism
$\mathbb{F}((t^{-1}))\cong\mathbb{F}[t]\oplus\left(\mathbb{F}((t^{-1}))/\mathbb{F}[t]\right)$.
The corresponding statement in the more familiar characteristic zero setting
is not true: $\mathbb{R}\not\cong\mathbb{Z}\oplus\mathbb{T}$.

To prove Theorem 1.6, it then suffices to show: for $\sigma$-a.e.
$x\in\mathbb{F}((t^{-1}))/\mathbb{F}[t]$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{e(P(n)x)}}=\begin{cases}1,&\text{if}~{}e(P(n)x)=1~{}\text{for
every}~{}n\in\mathbb{F}[t];\\\ 0,&\text{otherwise}.\end{cases}$

For $\mathbb{Z}$-action, total ergodicity is equivalent to the absence of
rational spectrum. Similarly, the assumption that $(T_{n})_{n\in\mathbb{N}}$
is a totally ergodic $\mathbb{F}[t]$-action means that

$\displaystyle\sigma\left(\left(\mathbb{F}(t)/\mathbb{F}[t]\right)\setminus\\{0\\}\right)=0$

Therefore, Theorem 1.6 reduces to studying equidistribution of the sequences
$\left(P(n)\alpha\right)_{n\in\mathbb{F}[t]}$ for irrational
$\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$.

A general Weyl-type equidistribution theorem for polynomials over
$\mathbb{F}[t]$ was established in [BL16], and we can use the result to finish
the proof of Theorem 1.6. First, we need some definitions for polynomials in
finite characteristic:

######  Definition 1.8.

  1. (1)

A polynomial $P(y)\in\mathbb{F}((t^{-1}))[y]$ is called _separable_ if
$P(y)=a_{0}+\sum_{i=1}^{k}{a_{i}y^{r_{i}}}$ and $p\nmid r_{i}$ for
$i\in\\{1,\dots,k\\}$.

  2. (2)

A polynomial $\eta(y)\in\mathbb{F}((t^{-1}))[y]$ is _additive_ if
$\eta(x+y)=\eta(x)+\eta(y)$ for any $x,y\in\mathbb{F}((t^{-1}))$.

  3. (3)

For $u\in\mathbb{F}((t^{-1}))$ and
$f:\mathbb{F}((t^{-1}))\to\mathbb{F}((t^{-1}))$, define the differencing
operator $\partial_{u}f(x)=f(x+u)-f(x)$. Then define
$\partial_{u_{1},\dots,u_{k}}$ recursively by
$\partial_{u_{1},\dots,u_{k}}=\partial_{u_{k}}\partial_{u_{1},\dots,u_{k-1}}$.
The _derivational degree_ (abbreviated d-$\deg$) of a polynomial
$P(y)\in\mathbb{F}((t^{-1}))[y]$ is the minimum $d\geq 0$ such that
$\partial_{u_{1},\dots,u_{d+1}}P(y)=0$ for any
$u_{1},\dots,u_{d+1},y\in\mathbb{F}((t^{-1}))$.

Note that d-$\deg{y^{p}}=1$, since $(u+v)^{p}=u^{p}+v^{p}$ in characteristic
$p$. More generally, for $r=\sum_{i=0}^{n}{a_{i}p^{i}}$ with
$a_{i}\in\\{0,\dots,p-1\\}$, we have d-$\deg{y^{r}}=\sum_{i=0}^{n}{a_{i}}$.
Any polynomial can be written in the form
$P(y)=a_{0}+\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$, where
$a_{0}\in\mathbb{F}((t^{-1}))$,
$\eta_{1},\dots,\eta_{n}\in\mathbb{F}((t^{-1}))[y]$ are additive polynomials,
and $y^{r_{i}}$ are distinct separable monomials.

We say that $a:\mathbb{F}_{p}[t]\to\mathbb{F}_{p}((t^{-1}))$ is _well-
distributed_ mod $\mathbb{F}_{p}[t]$ if

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{f(a(n))}}=\int_{\mathbb{F}_{p}((t^{-1}))/\mathbb{F}_{p}[t]}{f~{}dm}$

for every continuous function
$f:\mathbb{F}_{p}((t^{-1}))/\mathbb{F}_{p}[t]\to\mathbb{C}$ and every Følner
sequence $(\Phi_{N})_{N\in\mathbb{N}}$ in $\mathbb{F}_{p}[t]$. A more refined
notion of equidistribution is as follows. A function
$a:\mathbb{F}_{p}[t]\to\mathbb{F}_{p}((t^{-1}))$ is _well-distributed mod
$\mathbb{F}_{p}[t]$ in a subgroup
$H\subseteq\mathbb{F}_{p}((t^{-1}))/\mathbb{F}_{p}[t]$_ if

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{f(a(n))}}=\int_{H}{f~{}dm_{H}}$

for every continuous function
$f:\mathbb{F}_{p}((t^{-1}))/\mathbb{F}_{p}[t]\to\mathbb{C}$ and every Følner
sequence $(\Phi_{N})_{N\in\mathbb{N}}$ in $\mathbb{F}_{p}[t]$. For a subgroup
$H$ and a finite set $F\subseteq\mathbb{F}_{p}((t^{-1}))/\mathbb{F}_{p}[t]$,
we say that $a:\mathbb{F}_{p}[t]\to\mathbb{F}_{p}((t^{-1}))$ is _well-
distributed in the components of $H+F$_ if there exists
$m\in\mathbb{F}_{p}[t]\setminus\\{0\\}$ such that, for every
$k\in\mathbb{F}_{p}[t]$, the sequence
$\left(a(mn+k)\right)_{n\in\mathbb{F}_{p}[t]}$ is well-distributed in $H+x$
for some $x\in F$.

######  Theorem 1.9 ([BL16], Theorem 0.3).

An additive polynomial $\eta(y)\in(\mathbb{F}((t^{-1})))[y]$ is well
distributed in the subgroup555In [BL16], the subgroup $\mathcal{F}(\eta)$ is
called a _$\Phi$ -subtorus of level $\leq\log_{p}{d}$_
$\overline{\eta(\mathbb{F}[t])}=\mathcal{F}(\eta)+\eta(K)$, where
$K\subseteq\mathbb{F}((t^{-1}))/\mathbb{F}[t]$ is a finite subgroup. For any
polynomial $P(y)=\alpha_{0}+\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$, the orbit
closure $\mathcal{O}(P)=\overline{P(\mathbb{F}[t])}$ is of the form
$\mathcal{F}(P)+P(K)$, where
$\mathcal{F}(P)=\sum_{i=1}^{n}{\mathcal{F}(\eta_{i})}$ and $K$ is a finite
subset of $\mathbb{F}[t]$, and $P(y)$ is well-distributed in the components
$\mathcal{F}(P)+P(k)$, $k\in K$.

For an additive polynomial $\eta(y)\in\mathbb{F}[t]$ and irrational
$\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$, the orbit closure
$\overline{\left\\{\eta(y)\alpha:y\in\mathbb{F}[t]\right\\}}$ is equal to the
subtorus $\mathcal{F}(\eta\alpha)$ rather than a union of finitely many shifts
of $\mathcal{F}(\eta\alpha)$. It follows that for $P(y)\in\mathbb{F}[t]$ with
$P(0)=0$ and $\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$, the
sequence $\left(P(n)\alpha\right)_{n\in\mathbb{F}[t]}$ is well-distributed in
the subtorus $\mathcal{F}(P)$; see [BL16, Theorem 8.1] for more details. Thus,
for any $\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{e\left(P(n)\alpha\right)}}=\begin{cases}1,&\text{if}~{}e\left(P(n)\alpha\right)=1~{}\text{for
every}~{}n\in\mathbb{F}[t];\\\ 0,&\text{otherwise}.\end{cases}$

This completes the proof of Theorem 1.6.

We can now state our main result, which is an asymptotic version of Theorem
1.6 with quantitative bounds:

######  Theorem 1.10.

Let $P(y)\in(\mathbb{F}[t])[y]$ be a nonconstant polynomial of degree $d$ and
derivational degree $k$. Write
$P(y)=a_{0}+\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$. Let
$H_{i}=\eta_{i}\left(\mathbb{F}[t]\right)$ and $H=\sum_{i=1}^{n}{H_{i}}$. Then
for any $Q(t)\in\mathbb{F}[t]^{+}$ and any $f:\mathbb{F}[t]_{Q}\to\mathbb{C}$,

$\displaystyle\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{Q}}{f(x+a_{0}+z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q})}\leq\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}\right)^{1/2^{k-1}}\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q})},$

where $H_{Q}=\left\\{z\pmod{Q}:z\in H\right\\}$.

######  Remark 1.11.

(1) In the case $a_{0}=P(0)=0$, we get that the average
$\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}$
is approximated in $L^{2}(\mathbb{F}[t]_{Q})$ by the function

$\displaystyle\widetilde{f}(x)=\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{Q}}{f(x+z)},$

which is the projection of $f$ onto the space of $H_{Q}$-invariant functions,
so Theorem 1.10 can indeed be seen as a finitary version of Theorem 1.6.

(2) The phenomenon encompassed by Theorem 1.10 is simpler in the low degree
situation $(d<p)$. In the context of finite fields ($\mathbb{F}[t]_{Q}$ with
$Q$ irreducible in our notation), a closely related result was previously
established in [BBI21, Lemma 3]. In particular, it is shown that for any $q$,
any subsets $A,B\subseteq\mathbb{F}_{q}$, and any polynomial
$P(y)\in\mathbb{F}_{q}[y]$ with degree $d<p$,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x,y\in\mathbb{F}_{q}}{\mathbbm{1}_{A}(x)\mathbbm{1}_{B}(x+P(y))}-\frac{|A||B|}{q^{2}}\right|\leq\left(\frac{q^{d-1}-(q-1)^{d-1}}{q^{d-1}}\right)^{1/2^{d-1}}\ll
q^{-1/2^{d-1}};$

see the statement of Lemma 3 in [BBI21] and the formula for $\mathcal{E}(q,d)$
at the top of page 713.

As a consequence of Theorem 1.10, we obtain the following power savings for
the Furstenberg–Sárközy theorem:

######  Corollary 1.12.

Let $P(y)\in(\mathbb{F}[t])[y]$ be an intersective polynomial of degree $d$
and derivational degree $k$. Let $Q(t)\in\mathbb{F}[t]^{+}$. If
$A\subseteq\mathbb{F}[t]_{Q}$ does not contain distinct $a,b\in A$ with
$b-a=P(y)$ for some $y\in\mathbb{F}[t]_{Q}$, then

$\displaystyle|A|\ll_{P}|Q|\cdot\text{lpf}(Q)^{-1/2^{k-1}}.$

In particular, if $Q$ is irreducible (so that $\mathbb{F}[t]_{Q}$ is a field
with $|Q|$ elements), then

$\displaystyle|A|\ll_{P}|Q|^{1-1/2^{k-1}}.$

######  Remark 1.13.

(1) If we restrict to $\text{lpf}(Q)$ being sufficiently large (so that $P(y)$
does not reduce to the zero polynomial mod $Q$), then the implicit constant in
the conclusion of Corollary 1.12 depends only on the degree $d$ and the
derivational degree $k$.

(2) The bound given in Theorem 1.4 is difficult to compute in general and to
compare with the bound in Corollary 1.12. We can, however, highlight some
general features of the different bounds. The power savings obtained in
Corollary 1.12 depends only on the derivational degree $k$ of the polynomial
$P$ and applies to all intersective polynomials with coefficients in
$\mathbb{F}[t]$. By contrast, the Li–Sauermann bound depends on the degree $d$
of the polynomial $P$ and on the characteristic $p$ of the field
$\mathbb{F}_{q}$ and applies only to polynomials with zero constant term and
with coefficients in $\mathbb{F}_{p}$.

The disadvantage of our bound is that the power saving $\frac{1}{2^{k-1}}$
decays exponentially with the derivational degree. How the quantity $t_{p,d}$
appearing in Theorem 1.4 depends on $p$ and $d$ is not immediately clear from
the definition. However, a related bound due to Green [G17, Theorem 1.2]
(which Li and Sauermann optimize) gives power savings of

$\displaystyle
c^{\prime}(p,d)=\frac{1}{2d^{2}(p-1)^{2}(1+\log_{p}{d})^{2}\log{p}}$

for a degree $d$ polynomial $P(y)\in\mathbb{F}_{p}[y]$ with $P(0)=0$. That is,
for any $q=p^{k}$, the largest subset of $\mathbb{F}_{q}$ with no nontrivial
pattern $\\{x,x+P(y)\\}$ has cardinality

$\displaystyle|A|\ll_{p,d}q^{1-c^{\prime}(p,d)}.$

Therefore, for fixed $p$, there is a regime of sufficiently high degree
polynomials for which the Li–Sauermann bound beats ours. On the other hand,
the quantity $c^{\prime}(p,d)$ decays as $p\to\infty$, so the bound in
Corollary 1.12 will outperform this bound in sufficiently high characteristic
(for fixed degree $d$).

Thus, neither of the power saving bounds is universally better than the other,
and both methods have their advantages and disadvantages. The main goal of our
work is not to produce the best possible power saving bounds but to provide a
heuristic backing for why any power saving bound should hold at all and to
place the Furstenberg–Sárközy theorem over finite fields within the
appropriate general framework.

Combining Theorems 1.9 and 1.10, we may deduce several finitary combinatorial
statements from an infinitary statement about equidistribution. Say that a
polynomial $P(y)\in(\mathbb{F}[t])[y]$ is _good for irrational
equidistribution_ if $\left(P(n)\alpha\right)_{n\in\mathbb{F}[t]}$ is well
distributed for every irrational
$\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$.

######  Proposition 1.14.

A polynomial $P(y)$ is good for irrational equidistribution if and only if for
any totally ergodic system
$\left(X,\mathcal{X},\mu,(T_{n})_{n\in\mathbb{F}[t]}\right)$, any Følner
sequence $(\Phi_{N})_{N\in\mathbb{N}}$ in $\mathbb{F}[t]$, and any $f\in
L^{2}(\mu)$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{T_{P(n)}f}}=\int_{X}{f~{}d\mu}.$

Proposition 1.14 can be proved along the lines of Theorem 1.6 outlined earlier
using the spectral theorem for unitary actions of $\mathbb{F}[t]$. For the
“only if” direction, upon replacing $T_{n}$ by the multiplication operators
$(U_{n}h)(x)=e(nx)h(x)$ on
$L^{2}\left(\mathbb{F}((t^{-1}))/\mathbb{F}[t],\sigma\right)$, we use the fact
that $P(y)$ is good for irrational equidistribution to conclude

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{U_{P(n)}h(x)}}=\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{e(P(n)x)h(x)}}=\begin{cases}h(0),&\text{if}~{}x=0;\\\
0,&\text{otherwise}\end{cases}$

in $L^{2}(\sigma)$. This corresponds to the desired convergence result

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{T_{P(n)}f}}=\int_{X}{f~{}d\mu}.$

For the “if” direction: suppose $P(y)$ is not good for irrational
equidistribution, and let
$\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$ such that
$\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{e\left(P(y)\alpha\right)}}\neq
0$. We then take as our totally ergodic system
$X=\mathbb{F}((t^{-1}))/\mathbb{F}[t]$, $\mathcal{X}=Borel(X)$, $\mu=m_{X}$,
and $T_{n}x=x+n\alpha$. For the function $f(x)=e(x)$, we have
$\int_{X}{f(x)~{}dx}=0$, since $x\mapsto e(x)$ is a nontrivial character on
$X$, while

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{T_{P(n)}f}}=\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{e(P(n)\alpha)}}f\neq
0.$

######  Remark 1.15.

The naive analogue of Proposition 5.1 for $\mathbb{Z}$-actions is true. That
is, a polynomial $P(y)\in\mathbb{Z}[y]$ is good for irrational
equidistribution (meaning that $\left(P(n)\alpha\right)$ is well-distributed
mod 1 for every irrational $\alpha\in\mathbb{R}\setminus\mathbb{Q}$) if and
only if for any totally ergodic system $(X,\mathcal{X},\mu,T)$, any Følner
sequence $(\Phi_{N})_{N\in\mathbb{N}}$ in $\mathbb{Z}$, and any $f\in
L^{2}(\mu)$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{n\in\Phi_{N}}{T^{P(n)}f}}=\int_{X}{f~{}d\mu}.$

However, this result is far less meaningful in the setting of
$\mathbb{Z}$-actions, since every nonconstant integer polynomial is good for
irrational equidistribution by Weyl’s equidistribution theorem. This is far
from the case in the setting of $\mathbb{F}[t]$; see the examples below.

######  Example 1.16.

(1) Every nonconstant separable polynomial is good for irrational
equidistribution; see [BL16, Corollary 0.5].

(2) If $\eta_{1},\dots,\eta_{n}$ are additive polynomials such that
$\sum_{i=1}^{n}{c_{i}\eta_{i}(y)}=ay$ for some
$c_{1},\dots,c_{n}\in\mathbb{Z}$ and $a\in\mathbb{F}[t]$, then for any
distinct $r_{1},\dots,r_{n}\in\mathbb{N}$ not divisible by $p$,
$P(y)=\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$ is good for irrational
equidistribution. This follows from Theorem 1.9, since the condition
$\sum_{i=1}^{n}{c_{i}\eta_{i}(y)}=ay$ ensures that the orbit closure
$\sum_{i=1}^{n}{\mathcal{F}(\alpha\eta_{i})}$ contains the orbit
$\\{ay\alpha:y\in\mathbb{F}[t]\\}$, which is dense mod $\mathbb{F}[t]$ for
irrational $\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$.

(3) The polynomial $P(y)=y^{p^{2}}+y^{2p}-y$ is good for irrational
equidistribution. This follows from Theorem 1.18 below. Indeed, upon writing
$P(y)=\eta_{1}(y)+\eta_{2}(y^{2})$ with $\eta_{1}(y)=y^{p^{2}}-y$ and
$\eta_{2}(y)=y^{p}$, we see that $P(y)$ satisfies condition (iii) of Theorem
1.18 for $\zeta_{1}(y)=-y$ and $\zeta_{2}(y)=y^{p}$.

(4) The polynomial $P(y)=y^{p}$ is not good for irrational equidistribution:
for any $\alpha$ of the form $\alpha=\beta^{p}$, the orbit closure
$\overline{\left\\{P(y)\alpha:y\in\mathbb{F}[t]\right\\}}$ is contained in the
infinite index subgroup
$\\{x^{p}:x\in\mathbb{F}((t^{-1}))/\mathbb{F}[t]\\}\subseteq\mathbb{F}((t^{-1}))/\mathbb{F}[t]$.

(5) The polynomial $P(y)=y^{2p}-y^{2}$ is not good for irrational
equidistribution. Write $P(y)=\eta(y^{2})$ with $\eta(y)=y^{p}-y$. Then
clearly $P(\mathbb{F}[t])\subseteq\eta(\mathbb{F}[t])$. For any
$\alpha\in\mathbb{F}((t^{-1}))/\mathbb{F}[t]$ of the form

$\displaystyle\alpha=\sum_{j=0}^{\infty}{\alpha_{j}t^{-(j+1)}}$

satisfying $\alpha_{pj}=\alpha_{j}$ for $j\geq 0$, one can check by direct
calculation that $e\left(\eta(y)\alpha\right)=1$ for every
$y\in\mathbb{F}[t]$. Hence, for any such $\alpha$,
$\left(P(n)\alpha\right)_{n\in\mathbb{F}[t]}$ is not well distributed mod
$\mathbb{F}[t]$. Moreover, the set of all such $\alpha$ is uncountable so
contains irrational elements.

(6) An additive polynomial $P(y)=\sum_{i=0}^{k}{a_{i}y^{p^{i}}}$ is good for
irrational equidistribution if and only if $P(y)=a_{0}y$; see Proposition 5.4.

The following theorem summarizes the main achievements in this paper. In
particular, it emphasizes the role of equidistribution properties in obtaining
finitary combinatorial results over the rings $\mathbb{F}[t]_{Q}$. Note that
item (v) below strengthens the conclusion of Corollary 1.12 under the
assumption that the polynomial $P(y)$ is good for irrational equidistribution.

######  Theorem 1.17.

Let $P(y)\in(\mathbb{F}[t])[y]$ be a nonconstant polynomial. The following are
equivalent:

  1. (i)

for any $Q(t)\in\mathbb{F}[t]^{+}$,

$\displaystyle\sup_{\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q})}=1}{\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q})}}=o_{\text{lpf}(Q)\to\infty}(1);$

  2. (ii)

there exist $C_{1},C_{2},\gamma>0$ such that for any
$Q(t)\in\mathbb{F}[t]^{+}$ with $\text{lpf}(Q)\geq C_{1}$, one has

$\displaystyle\sup_{\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q})}=1}{\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q})}}\leq
C_{2}\cdot\text{lpf}(Q)^{-\gamma};$

  3. (iii)

there exists $C>0$ such that if $Q(t)\in\mathbb{F}[t]^{+}$ and
$\text{lpf}(Q)\geq C$, then $H+Q\mathbb{F}[t]=\mathbb{F}[t]$, where
$H\leq(\mathbb{F}[t],+)$ is the group generated by
$\\{P(y)-P(0):y\in\mathbb{F}[t]\\}$.

  4. (iv)

for any $\delta>0$, there exists $N>0$ such that if $Q(t)\in\mathbb{F}[t]^{+}$
has $\text{lpf}(Q)\geq N$ and $A,B\subseteq\mathbb{F}[t]_{Q}$ are subsets with
$|A||B|\geq\delta|Q|^{2}$, then there exist $x,y\in\mathbb{F}[t]_{Q}$ such
that $x\in A$ and $x+P(y)\in B$;

  5. (v)

there exist $C_{1},C_{2},\gamma>0$ such that for any
$Q(t)\in\mathbb{F}[t]^{+}$ with $\text{lpf}(Q)\geq C_{1}$, one has

$\displaystyle\bigg{|}\left|\left\\{(x,y)\in\mathbb{F}[t]_{Q}^{2}:x\in
A,x+P(y)\in B\right\\}\right|-|A||B|\bigg{|}\leq
C_{2}|A|^{1/2}|B|^{1/2}|Q|\cdot\text{lpf}(Q)^{-\gamma}.$

Moreover, if $P(y)$ is good for irrational equidistribution, then each of the
properties (i)-(v) holds.

By restricting the coefficients of the polynomial $P$, we can prove a stronger
version of Theorem 1.17:

######  Theorem 1.18.

Let $P(y)\in\mathbb{F}_{p}[y]$. Let
$\eta_{1},\dots,\eta_{n}\in\mathbb{F}_{p}[y]$ be additive polynomials and
$r_{1},\dots,r_{n}\in\mathbb{N}$ distinct positive integers not divisible by
$p$ so that $P(y)=\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$. The following are
equivalent:

  1. (i)

$P(y)$ is good for irrational equidistribution;

  2. (ii)

for any totally ergodic system
$\left(X,\mathcal{X},\mu,(T_{y})_{y\in\mathbb{F}[t]}\right)$, any Følner
sequence $(\Phi_{N})_{N\in\mathbb{N}}$ in $\mathbb{F}[t]$, and any $f\in
L^{2}(\mu)$,

$\displaystyle\lim_{N\to\infty}{\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\Phi_{N}}{T_{P(y)}f}}=\int_{X}{f~{}d\mu};$

  3. (iii)

there exist additive polynomials
$\zeta_{1},\dots,\zeta_{n}\in\mathbb{F}_{p}[y]$ and
$a\in\mathbb{F}_{p}^{\times}$ such that

$\displaystyle\sum_{i=1}^{n}{(\eta_{i}\circ\zeta_{i})(y)}=ay;$

  4. (iv)

for any $Q(t)\in\mathbb{F}[t]^{+}$,

$\displaystyle\sup_{\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q})}=1}{\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q})}}=o_{\text{lpf}(Q)\to\infty}(1);$

  5. (v)

there exist $C_{1},C_{2},\gamma>0$ such that for any
$Q(t)\in\mathbb{F}[t]^{+}$ with $\text{lpf}(Q)\geq C_{1}$, one has

$\displaystyle\sup_{\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q})}=1}{\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q})}}\leq
C_{2}\cdot\text{lpf}(Q)^{-\gamma};$

  6. (vi)

there exists $C>0$ such that if $Q(t)\in\mathbb{F}[t]^{+}$ and
$\text{lpf}(Q)\geq C$, then $H_{Q}=\mathbb{F}[t]$, where
$H_{Q}=\sum_{i=1}^{n}{H_{i,Q}}$, $H_{i,Q}=\eta_{i}(\mathbb{F}[t]_{Q})$.

  7. (vii)

for any $\delta>0$, there exists $N>0$ such that if $Q(t)\in\mathbb{F}[t]^{+}$
has $\text{lpf}(Q)\geq N$ and $A,B\subseteq\mathbb{F}[t]_{Q}$ are subsets with
$|A||B|\geq\delta|Q|^{2}$, then there exist $x,y\in\mathbb{F}[t]_{Q}$ such
that $x\in A$ and $x+P(y)\in B$;

  8. (viii)

there exist $C_{1},C_{2},\gamma>0$ such that for any
$Q(t)\in\mathbb{F}[t]^{+}$ with $\text{lpf}(Q)\geq C_{1}$, one has

$\displaystyle\bigg{|}\left|\left\\{(x,y)\in\mathbb{F}[t]_{Q}^{2}:x\in
A,x+P(y)\in B\right\\}\right|-|A||B|\bigg{|}\leq
C_{2}|A|^{1/2}|B|^{1/2}|Q|\cdot\text{lpf}(Q)^{-\gamma}.$

Combining Furstenberg–Sárközy-type results with the technology of Loeb
measures on ultraproducts, we are able to establish partition regularity of
families of polynomial equations over finite fields, such as the following:

######  Theorem 1.19.

Let $P(x)\in\mathbb{F}_{p}[x]$ be a nonconstant polynomial, and let
$Q(x)\in\mathbb{F}_{p}[x]$ be good for irrational equidistribution. For any
$r\in\mathbb{N}$, there exists $K=K(P,Q,r)\in\mathbb{N}$ and $c=c(P,Q,r)>0$
such that for any $k\geq K$ and any $r$-coloring
$\mathbb{F}_{p^{k}}=\bigcup_{i=1}^{r}{C_{i}}$, there are at least
$c(p^{k})^{2}$ monochromatic solutions to the equation $P(x)-P(y)=Q(z)$. That
is,

$\displaystyle\left|\left\\{(x,y,z)\in\mathbb{F}_{p^{k}}^{3}:P(x)-P(y)=Q(z)~{}\text{and}~{}\\{x,y,z\\}\subseteq
C_{i}~{}\text{for some}~{}i\in\\{1,\dots,r\\}\right\\}\right|\geq
c(p^{k})^{2}.$

We are in fact able to prove partition regularity under a weaker (but more
technically cumbersome) condition on $Q$; see Corollary 6.7. Under this weaker
assumption, one application of note is a polynomial Schur theorem over finite
fields:

######  Corollary 1.20.

Let $P(x)\in\mathbb{Z}[x]$ with $P(0)=0$. Then for any $r\in\mathbb{N}$, there
exists $N=N(P,r)\in\mathbb{N}$ and $c=c(P,r)>0$ such that if $q>N$ and
$\mathbb{F}_{q}=\bigcup_{i=1}^{r}{C_{i}}$, then there are at least $cq^{2}$
monochromatic solutions to the equation $P(x)+P(y)=P(z)$. In particular, if
the coefficients of $P$ are not all divisible by the characteristic of
$\mathbb{F}_{q}$, then there are $\gg_{P,r}q^{2}$ monochromatic solutions with
$P(x),P(y),P(z)\neq 0$.

######  Remark 1.21.

In the case $P(x)=x^{d}$, Corollary 1.20 corresponds to the Fermat equation
$x^{d}+y^{d}=z^{d}$. Schur proved the existence of solutions to the Fermat
equation in all prime fields of sufficiently high characteristic using his
eponymous partition regularity theorem in [S16]. (This was in fact the
original motivation for Schur’s theorem.) The much stronger property of
partition regularity of the Fermat equation was established previously in the
context of prime fields in [CGS12, Theorem 4] and generalized to a family of
related polynomial equations in [L18]. We complete the picture here by
extending the partition regularity property to arbitrary finite fields of
sufficiently large order (with no assumption on the characteristic).

Related density results for Pythagorean pairs and triples in finite fields
were obtained in [DLMS23, Section 6], where they also show that a density
version (“density regularity”) of Corollary 1.20 fails already for the
Pythagorean equation $x^{2}+y^{2}=z^{2}$.

The paper is organized as follows. In Section 2, we prove Theorem 1.5, showing
that $\text{lpf}(Q)$ appropriately captures the phenomenon of asymptotic total
ergodicity. Section 3 is dedicated to the proof of Theorem 1.10. The final
three sections concern applications of Theorem 1.10: Section 4 deals with
power saving bounds in the Furstenberg–Sárközy theorem for intersective
polynomials in the presence of asymptotic total ergodicity (Corollary 1.12);
Section 5 with further enhancements of the Furstenberg–Sárközy theorem for
polynomials with good equidistributional behavior (Theorems 1.17 and 1.18);
and Section 6 with partition regularity of polynomial equations over finite
fields. The relevant tools are introduced in the corresponding sections as
needed.

##  2\. Asymptotic total ergodicity

In this section, we prove that the quantity $\text{lpf}(Q)$ captures the
phenomenon of asymptotic total ergodicity. Recall Theorem 1.5:

See 1.5

###### Proof.

(ii) $\implies$ (i). Fix $m\in\mathbb{F}[t]\setminus\\{0\\}$. If
$Q\in\mathbb{F}[t]^{+}$ has $\text{lpf}(Q)>|m|$, then $m$ is an element of the
multiplicative group $\mathbb{F}[t]_{Q}^{\times}$. Hence, for any
$x\in\mathbb{F}[t]_{Q}$ and any $f:\mathbb{F}[t]_{Q}\to\mathbb{C}$,

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+my)}=\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}.$

(i) $\implies$ (ii). Let $Q\in\mathbb{F}[t]^{+}$, and let $m$ be an
irreducible factor of $Q$. Enumerate
$\mathbb{F}[t]_{m}=\\{x_{0},\dots,x_{|m|-1}\\}$, and let
$f_{0}:\mathbb{F}[t]_{m}\to\mathbb{D}$ be the function $f_{0}(x_{k})=e^{2\pi
ik/|m|}$. Define $f:\mathbb{F}[t]_{Q}\to\mathbb{D}$ by $f(x)=f_{0}(x\mod{m})$.
Then

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+my)}=f(x)$

for every $x\in\mathbb{F}[t]_{Q}$. On the other hand,

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}=\operatorname*{\text{\Large$\mathbb{E}$}}_{k=0}^{|m|-1}{\operatorname*{\text{\Large$\mathbb{E}$}}_{z\equiv
x_{k}}{f(z)}}=\operatorname*{\text{\Large$\mathbb{E}$}}_{k=0}^{|m|-1}{f_{0}(x_{k})}=0.$

Therefore,

$\displaystyle\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+my)}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q}}{f(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q})}^{2}=\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q}}{\left|f(x)\right|^{2}}=1.$

Now suppose $\text{lpf}(Q_{n})\not\to\infty$. Taking a subsequence if
necessary, we may assume $\text{lpf}(Q_{n})$ is bounded. By the pigeonhole
principle, we may then take a further subsequence and assume that there is a
common irreducible factor $m$ of every $Q_{n}$, $n\in\mathbb{N}$. The above
calculation shows that we may find $f_{n}:\mathbb{F}[t]_{Q_{n}}\to\mathbb{D}$
with

$\displaystyle\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q_{n}}}{f_{n}(x+my)}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}[t]_{Q_{n}}}{f_{n}(z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q_{n}})}=1$

for each $n\in\mathbb{N}$, contradicting (i). ∎

##  3\. Asymptotic projection theorem

We now turn to proving Theorem 1.10 with Fourier analysis. Characters on
$\mathbb{F}[t]_{Q}$ take the form $\chi(x)=e(sx/Q)$ for some
$s\in\mathbb{F}[t]_{Q}$. For a function $f:\mathbb{F}[t]_{Q}\to\mathbb{C}$, we
therefore define its Fourier transform
$\widehat{f}:\mathbb{F}[t]_{Q}\to\mathbb{C}$ by

$\displaystyle\widehat{f}(s)=\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q}}{f(x)e(-sx/Q)}.$

We state some basic properties of the Fourier transform

######  Proposition 3.1.

For any $f:\mathbb{F}\to\mathbb{C}$, one has

  * •

Fourier inversion:

$\displaystyle f(x)=\sum_{s\in\mathbb{F}[t]_{Q}}{\widehat{f}(s)e(sx/Q)}.$

  * •

Parseval’s identity:

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q}}{\left|f(x)\right|^{2}}=\sum_{s\in\mathbb{F}[t]_{Q}}{\left|\widehat{f}(s)\right|^{2}}.$

Define
$F(x):=\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{Q}}{f(x+a_{0}+z)}$. Then

$\displaystyle\widehat{F}(s)=\left(\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(sP(y)/Q)}-e(sa_{0}/Q)\mathbbm{1}_{H_{Q}^{\perp}}(s)\right)\widehat{f}(s),$

where $H_{Q}^{\perp}=\left\\{s\in\mathbb{F}[t]_{Q}:e(sz)=1~{}\text{for
all}~{}z\in H_{Q}\right\\}$ is the annihilator of the subgroup $H_{Q}$. Hence,
by Parseval’s identity

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q}}{\left|F(x)\right|^{2}}$
$\displaystyle=\sum_{s\in\mathbb{F}[t]_{Q}}{\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(sP(y)/Q)}-e(sa_{0}/Q)\mathbbm{1}_{H_{Q}^{\perp}}(s)\right|^{2}\left|\widehat{f}(s)\right|^{2}}$
$\displaystyle\leq\left(\sup_{s\in\mathbb{F}[t]_{Q}}{\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(sP(y)/Q)}-e(sa_{0}/Q)\mathbbm{1}_{H_{Q}^{\perp}}(s)\right|}\right)^{2}\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q}}{|f(x)|^{2}}.$

All that remains to show is the inequality:

######  Proposition 3.2.

Let $P(y)\in(\mathbb{F}[t])[y]$ be a nonconstant polynomial of degree $d$ and
derivational degree $k$. Write
$P(y)=a_{0}+\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$. Then for any
$Q(t)\in\mathbb{F}[t]^{+}$ and any $s\in\mathbb{F}[t]_{Q}$,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(sP(y)/Q)}-e(sa_{0}/Q)\mathbbm{1}_{H_{Q}^{\perp}}(s)\right|^{2^{k-1}}\leq
p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}.$

A key ingredient in Proposition 3.2 is the following van der Corput-type
inequality. We do not use any ring structure for this result, so we state and
prove it in the setting of an arbitrary finite abelian group $G$. For a
function $f:G\to\mathbb{C}$, define a multiplicative differencing operator by
$\Delta_{v}f(u)=f(u+v)\overline{f(u)}$, and let
$\Delta_{v_{1},\dots,v_{k}}f=\Delta_{v_{k}}\left(\Delta_{v_{1},\dots,v_{k-1}}f\right)$
for $k\in\mathbb{N}$ and $v_{1},\dots,v_{k}\in G$.

######  Lemma 3.3.

Let $G$ be a finite abelian group, and let $H\leq G$ be a subgroup. For any
function $f:G\to\mathbb{C}$ and any $k\in\mathbb{N}$,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}\right|^{2^{k}}\leq\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{1},\dots,v_{k}\in
H}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{1},\dots,v_{k}}f(u)}}$

######  Remark 3.4.

It is worth commenting on two special cases of Lemma 3.3. When $H=G$, the
quantity on the right hand side is equal to
$\left\|f\right\|_{U^{k}(G)}^{2^{k}}$, so the conclusion of Lemma 3.3 reduces
to the inequality
$\left\|f\right\|_{U^{1}(G)}\leq\left\|f\right\|_{U^{k}(G)}$, which is a
special case of monotonicity for the Gowers (semi)norms. On the other hand,
when $H=\\{0\\}$, the right hand side is equal to
$\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in G}{|f(u)|^{2^{k}}}$, so the
conclusion of Lemma 3.3 follows by Jensen’s inequality. The general case can
be seen as interpolating between these two extremes.

###### Proof.

Suppose $k=1$. Note that

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}=\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{\operatorname*{\text{\Large$\mathbb{E}$}}_{h\in H}{f(x+h)}}.$

Therefore, by Jensen’s inequality,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}\right|^{2}=\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{\operatorname*{\text{\Large$\mathbb{E}$}}_{h\in
H}{f(x+h)}}\right|^{2}\leq\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{h\in
H}{f(x+h)}\right|^{2}}=\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{\operatorname*{\text{\Large$\mathbb{E}$}}_{h_{1},h_{2}\in
H}{f(x+h_{1})\overline{f(x+h_{2})}}}.$

Interchanging the order of averaging and making the substitutions
$v=h_{1}-h_{2}$, $u=x-h_{2}$, we obtain the desired inequality

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}\right|^{2}\leq\operatorname*{\text{\Large$\mathbb{E}$}}_{v\in
H}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{f(u+v)\overline{f(u)}}}.$

Suppose the inequality holds for $k-1$. Then

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}\right|^{2^{k}}=\left(\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}\right|^{2^{k-1}}\right)^{2}\leq\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{1},\dots,v_{k-1}\in
H}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{1},\dots,v_{k-1}}f(u)}}\right|^{2},$

which is in turn bounded above by

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{1},\dots,v_{k-1}\in
H}{\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{1},\dots,v_{k-1}}f(u)}\right|^{2}}.$

For fixed $v_{1},\dots,v_{k-1}$, applying the $k=1$ case with the function
$\Delta_{v_{1},\dots,v_{k-1}}f$ gives

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{1},\dots,v_{k-1}}f(u)}\right|^{2}\leq\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{k}\in
H}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{k}}\Delta_{v_{1},\dots,v_{k-1}}f(x)}}=\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{k}\in
H}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{1},\dots,v_{k-1},v_{k}}f(x)}}.$

Putting everything together,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in
G}{f(x)}\right|^{2^{k}}\leq\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{1},\dots,v_{k}\in
H}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in
G}{\Delta_{v_{1},\dots,v_{k}}f(u)}}.$

∎

###### Proof of Proposition 3.2.

We first make a reduction to separable polynomials. If $s\in H_{Q}^{\perp}$,
then $e(sP(y)/Q)=e(sa_{0}/Q)e(s(P(y)-a_{0})/Q)=e(sa_{0}/Q)$ for every
$y\in\mathbb{F}[t]_{Q}$, since $P(y)-a_{0}\in H_{Q}$. Suppose now that
$s\notin H_{Q}^{\perp}$. We want to show

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e\left(sP(y)/Q\right)}\right|^{2^{k-1}}\leq
p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}.$

Noting that $H_{Q}^{\perp}=\cap_{i=1}^{n}{H_{i,Q}^{\perp}}$, we have

$\displaystyle I=\left\\{1\leq i\leq n:s\notin
H_{i,Q}^{\perp}\right\\}\neq\emptyset.$

Moreover, for any $y\in\mathbb{F}[t]_{Q}$,

$\displaystyle e\left(sP(y)/Q\right)=e(sa_{0}/Q)e\left(s\sum_{i\in
I}{\eta_{i}(y^{r_{i}})}/Q\right).$

Now, for each $i\in I$, the function $\chi_{i}(x)=e(s\eta_{i}(x)/Q)$ is a
nontrivial character on $\mathbb{F}[t]_{Q}$, so there exists $s_{i}\neq 0$
such that $\chi_{i}(x)=e(s_{i}x/Q)$. It therefore suffices to prove the
following: for any nonconstant separable polynomial
$P(y)\in(\mathbb{F}[t]_{Q})[y]$,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e\left(P(y)/Q\right)}\right|^{2^{k-1}}\leq
p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}.$

Suppose $k=1$. Then $P(y)=s_{1}y+s_{0}\in(\mathbb{F}[t]_{Q})[y]$ with
$s_{1}\neq 0$. Therefore,

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(P(y)/Q)}=e(s_{0}/Q)\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(s_{1}y/Q)}=0.$

Now suppose $k\geq 2$. Let
$P(y)=\sum_{i=i}^{n}{s_{i}y^{r_{i}}}+P^{\prime}(y)$, where
d-$\deg{y^{r_{i}}}=k$ and d-$\deg{P^{\prime}(y)}\leq k-1$. By Lemma 3.3,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(P(y)/Q)}\right|^{2^{k-1}}\leq\operatorname*{\text{\Large$\mathbb{E}$}}_{v_{1},\dots,v_{k-1}\in
K}{\operatorname*{\text{\Large$\mathbb{E}$}}_{u\in\mathbb{F}[t]_{Q}}{e\left(\partial_{v_{1},\dots,v_{k-1}}P(u)/Q\right)}}$

for any subgroup $K\leq(\mathbb{F}[t]_{Q},+)$. (We will take a convenient
choice for $K$ later.) We now wish to obtain an expression for
$\partial_{v_{1},\dots,v_{k}}P(u)$ that will allow us to bound the avaerage

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{u}{e\left(s\partial_{v_{1},\dots,v_{k-1}}P(u)/Q\right)}.$

For $v_{1},\dots,v_{k-1}\in\mathbb{F}[t]_{Q}$, one has that
$\partial_{v_{1},\dots,v_{k-1}}P^{\prime}(u)$ is constant (as a function of
$u$), since d-$\deg{P^{\prime}(y)}\leq k-1$, so we can pull the constant
$e\left(s\partial_{v_{1},\dots,v_{k-1}}P^{\prime}(u)\right)$ outside of the
average.

Let $m=\left\lfloor\log_{p}{d}\right\rfloor$ so that $p^{m}\leq d$ and
$p^{m+1}>d$. For each $i=1,\dots,n$, we may write
$r_{i}=\sum_{j=0}^{m}{c_{i,j}p^{j}}$ with $c_{i,j}\in\\{0,\dots,p-1\\}$ and
$\sum_{j=0}^{m}{c_{i,j}}=k$. Since $P$ is separable by assumption, we have
$c_{i,0}\neq 0$ for $i\in\\{1,\dots,n\\}$. Then

$\displaystyle\partial_{v_{1},\dots,v_{k-1}}(y^{r_{i}})=b_{i}\sum_{j=1}^{m}{S_{i,j}(v_{1},\dots,v_{k-1})u^{p^{j}}}+R_{i}(v_{1},\dots,v_{k-1}),$

where $b_{i}=\prod_{j=1}^{m}{c_{i,j}!}$, $S_{i,j}(v_{1},\dots,v_{k-1})$ is the
sum of all monomials of the form

$\displaystyle\prod_{l=1}^{k-1}{v_{l}^{p^{j_{l}}}}$

with

$\displaystyle\left|\left\\{1\leq l\leq
k-1:j_{l}=j^{\prime}\right\\}\right|=\begin{cases}c_{i,j^{\prime}},&\text{if}~{}j^{\prime}\neq
j\\\ c_{i,j}-1,&\text{if}~{}j^{\prime}=j\end{cases}$

and $R_{i}$ is a symmetric polynomial in $k-1$ variables. (If $c_{i,j}=0$,
then $S_{i,j}=0$.) We can therefore write

$\displaystyle\partial_{v_{1},\dots,v_{k-1}}P(y)$
$\displaystyle=\sum_{i=1}^{n}{s_{i}b_{i}\sum_{j=1}^{m}{S_{i,j}(v_{1},\dots,v_{k-1})u^{p^{j}}}}+R(v_{1},\dots,v_{k-1})$
$\displaystyle=\sum_{j=0}^{m}{\left(\sum_{i=1}^{n}{s_{i}b_{i}S_{i,j}(v_{1},\dots,v_{k-1})}\right)u^{p^{j}}}+R(v_{1},\dots,v_{k-1}),$

where $R=\sum_{i=1}^{n}{R_{i}}$.

Let

$\displaystyle\eta_{v_{1},\dots,v_{k-1}}(u)=\sum_{j=0}^{m}{\left(\sum_{i=1}^{n}{s_{i}b_{i}S_{i,j}(v_{1},\dots,v_{k-1})}\right)u^{p^{j}}}.$

Note that $\eta_{v_{1},\dots,v_{k-1}}$ is a group homomorphism
$(\mathbb{F}[t]_{Q},+)\to(\mathbb{F}[t]_{Q},+)$. It follows that

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{u}{e\left(\partial_{v_{1},\dots,v_{k-1}}P(u)/Q\right)}\right|=\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{u}{e\left(\eta_{v_{1},\dots,v_{k-1}}(u)/Q\right)}\right|=0$

whenever $e\left(\eta_{v_{1},\dots,v_{k-1}}(\cdot)/Q\right)$ is a nonzero
function. Noting that $e\left(\eta_{v_{1},\dots,v_{k-1}}(\cdot)/Q\right)$ is a
character on $\mathbb{F}[t]_{Q}$, it may be written in the form
$e\left(\varphi(v_{1},\dots,v_{k-1})u/Q\right)$ for some
$\varphi(v_{1},\dots,v_{k-1})\in\mathbb{F}[t]_{Q}$. We have thus obtained the
bound

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(P(y)/Q)}\right|^{2^{k-1}}\leq\frac{\left|\left\\{(v_{1},\dots,v_{k-1})\in
K^{k-1}:\varphi(v_{1},\dots,v_{k-1})=0\right\\}\right|}{|K|^{k-1}}.$

The remainder of the proof consists of two main steps. First, we show that,
for a convenient choice of $K$, the function $\varphi$ becomes (after a change
of coordinates) a polynomial in $k-1$ variables. Next, we establish a bound on
the number of roots of multivariable polynomials mod $Q$.

Recall $m=\left\lfloor\log_{p}{d}\right\rfloor$. Let
$K=\\{x^{p^{m}}:x\in\mathbb{F}[t]_{Q}\\}$. This is a subgroup, since the
function $x\mapsto x^{p^{m}}$ is a homomorphism. For $0\leq j\leq m$, let
$C_{j}=\sum_{i=1}^{n}{s_{i}b_{i}S_{i,j}}$ so that

$\displaystyle\eta_{v_{1},\dots,v_{k-1}}(u)=\sum_{j=0}^{m}{C_{j}(v_{1},\dots,v_{k-1})u^{p^{j}}},$

and each of the polynomials $C_{j}(v_{1},\dots,v_{k-1})$ is an additive
polynomial of degree at most $p^{m}$ in each coordinate. In particular,
$v_{i}\mid C_{j}(v_{1},\dots,v_{k-1})$ for each $i\in\\{1,\dots,k-1\\}$.
Making the substitution $v_{i}=w_{i}^{p^{m}}$, we therefore have

$\displaystyle\eta_{w_{1}^{p^{m}},\dots,w_{k-1}^{p^{m}}}(u)=\sum_{j=0}^{m}{C_{j}\left(w_{1}^{p^{m}},\dots,w_{k-1}^{p^{m}}\right)u^{p^{j}}}=\sum_{j=0}^{m}{\widetilde{C}_{j}\left(w_{1},\dots,w_{k-1}\right)w_{1}^{p^{j}}\dots
w_{k-1}^{p^{j}}u^{p^{j}}}$

for some $\widetilde{C}_{j}$.

For each $j\geq 0$, the function $\chi(u)=e(u^{p^{j}}/Q)$ is a character on
$\mathbb{F}[t]_{Q}$, so there exists $z_{j}\in\mathbb{F}[t]_{Q}$ such that
$\chi(u)=e(z_{j}u/Q)$. Hence, defining
$\psi(w_{1},\dots,w_{k-1})=\varphi\left(w_{1}^{p^{m}},\dots,w_{k-1}^{p^{m}}\right)$,
we have

$\displaystyle\psi(w_{1},\dots,w_{k-1})=\sum_{j=0}^{m}{z_{j}\widetilde{C}_{j}(w_{1},\dots,w_{k-1})w_{1}\dots
w_{k-1}}.$

That is, $\psi$ is a polynomial of degree at most $p^{2m}$ in each coordinate.

We claim that $\psi$ is not the zero polynomial. By definition, $z_{0}=1$. We
also have

$\displaystyle
C_{0}(v_{1},\dots,v_{k-1})=\sum_{i=1}^{n}{s_{i}b_{i}S_{i,0}(v_{1},\dots,v_{k-1})}.$

The coefficients $b_{i}$ are integers coprime to $p$, so
$b_{i}\in\mathbb{F}[t]_{Q}^{\times}$. Hence, $s_{i}b_{i}\neq 0$. Now,
$S_{i,0}$ is a sum of terms of the form

$\displaystyle\prod_{l=1}^{k-1}{v_{l}^{p^{j}_{l}}}$

with the property $\sum_{l=1}^{k-1}{p^{j_{l}}}=r_{i}-p$. Therefore, the
monomials appearing in $S_{i,0}$ are distinct from the monomials appearing in
$S_{i^{\prime},0}$ for $i\neq i^{\prime}$. It follows that $C_{0}$ is not the
zero polynomial. Thus,

$\displaystyle z_{0}\widetilde{C}_{0}(w_{1},\dots,w_{k-1})w_{1}\dots
w_{k-1}=C_{0}\left(w_{1}^{p^{m}},\dots,w_{k-1}^{p^{m}}\right)$

is not the zero polynomial, and each monomial appearing has degree divisible
by $p^{m}$. Finally, for $j\neq 0$, we have

$\displaystyle\widetilde{C}_{j}(w_{1},\dots,w_{k-1})w_{1}\dots
w_{k-1}=\frac{C_{j}\left(w_{1}^{p^{m}},\dots,w_{k-1}^{p^{m}}\right)}{\prod_{l=1}^{k-1}{w_{l}^{p^{j}-1}}},$

which consists of monomials in which each variable has degree congruent to $1$
mod $p$. This proves that $\psi$ is not the zero polynomial.

The final step is to show that $\psi$ has only a small number of zeros.

######  Lemma 3.5.

Let $l\in\mathbb{N}$, and let
$T(y_{1},\dots,y_{l})\in(\mathbb{F}[t]_{Q})[y_{1},\dots,y_{l}]$ be a nonzero
polynomial of degree $d_{i}$ in the variable $y_{i}$ for $i=1,\dots,d$. Then

$\displaystyle\left|\left\\{(y_{1},\dots,y_{l})\in\mathbb{F}[t]_{Q}^{l}:T(y_{1},\dots,y_{l})=0\right\\}\right|\leq\left(\sum_{i=1}^{l}{d_{i}}\right)\frac{|Q|^{l}}{\text{lpf}(Q)}.$

###### Proof of Lemma.

Let us first consider the case $l=1$. Write
$T(y)=\alpha_{d}y^{d}+\dots+\alpha_{1}y+\alpha_{0}$. We view
$\alpha_{0},\dots,\alpha_{d}$ as elements of $\mathbb{F}[t]$ with
$|\alpha_{i}|<|Q|$. Let $\alpha=\gcd(\alpha_{0},\dots,\alpha_{d},Q)$,
$\widetilde{\alpha}_{i}=\frac{\alpha_{i}}{\alpha}$, and
$\widetilde{T}(y)=\widetilde{\alpha}_{d}y^{d}+\dots+\widetilde{\alpha}_{1}y+\widetilde{\alpha}_{0}$.
Fix $Q^{\prime}\in\mathbb{F}[t]^{+}$ irreducible such that
$Q^{\prime}\mid\frac{Q}{\alpha}$. For some $i\in\\{0,\dots,d\\}$, we have
$Q^{\prime}\nmid\widetilde{\alpha}_{i}$. Since $\mathbb{F}[t]_{Q^{\prime}}$ is
a field, and $\widetilde{T}$ reduces to a nonzero polynomial of degree $\leq
d$ mod $Q^{\prime}$, we have

$\displaystyle\left|\\{y^{\prime}\in\mathbb{F}[t]_{Q^{\prime}}:\widetilde{T}(y^{\prime})\equiv
0\pmod{Q^{\prime}}\\}\right|\leq d.$

Now suppose $T(y)=0$. Then $\alpha\widetilde{T}(y)=0$. That is,
$Q\mid\alpha\widetilde{T}(y)$, so $\frac{Q}{\alpha}\mid\widetilde{T}(y)$.
Hence, $Q^{\prime}\mid\widetilde{T}(y)$. Equivalently, $\widetilde{T}(y)\equiv
0\pmod{Q^{\prime}}$. Therefore,

$\displaystyle\left|\\{y\in\mathbb{F}[t]_{Q}:T(y)=0\\}\right|\leq
d\frac{|Q|}{|Q^{\prime}|}\leq d\frac{|Q|}{\text{lpf}(Q)}.$

Suppose $l\geq 2$. If $\text{lpf}(Q)\leq\sum_{i=1}^{l}{d_{i}}$, then there is
nothing to prove, so assume $\text{lpf}(Q)>\sum_{i=1}^{l}{d_{i}}$. Fix
$y\in\mathbb{F}[t]_{Q}$, and let
$T_{y}(y_{1},\dots,y_{l-1})=T(y_{1},\dots,y_{l-1},y)$. If
$T_{y}(y_{1},\dots,y_{l-1})\in(\mathbb{F}[t]_{Q})[y_{1},\dots,y_{l-1}]$ is not
the zero polynomial, then by the induction hypothesis,

$\displaystyle\left|\left\\{(y_{1},\dots,y_{l-1})\in\mathbb{F}[t]_{Q}^{l-1}:T_{y}(y_{1},\dots,y_{l-1})=0\right\\}\right|\leq\left(\sum_{i=1}^{l-1}{d_{i}}\right)\frac{|Q|^{l-1}}{\text{lpf}(Q)}.$

Hence,

$\displaystyle\left|\left\\{(y_{1},\dots,y_{l})\in\mathbb{F}[t]_{Q}^{l}:T(y_{1},\dots,y_{l})=0\right\\}\right|$
$\displaystyle~{}=|Q|^{l-1}\left|\\{y\in\mathbb{F}[t]_{Q}:T_{y}=0\\}\right|+\left|\\{(y_{1},\dots,y_{l-1},y):T_{y}\neq
0,T_{y}(y_{1},\dots,y_{l-1})=0\\}\right|$
$\displaystyle~{}\leq|Q|^{l-1}\left|\\{y\in\mathbb{F}[t]_{Q}:T_{y}=0\\}\right|+\left(\sum_{i=1}^{l-1}{d_{i}}\right)\frac{|Q|^{l}}{\text{lpf}(Q)}.$

It therefore suffices to prove

$\displaystyle\left|\\{y\in\mathbb{F}[t]_{Q}:T_{y}=0\\}\right|\leq
d_{l}\frac{|Q|}{\text{lpf}(Q)}.$

Fix $y_{1},\dots,y_{l-1}\in\mathbb{F}[t]_{Q}$, and let
$T^{y_{1},\dots,y_{l-1}}(y)=T_{y}(y_{1},\dots,y_{l-1})=T(y_{1},\dots,y_{l-1},y)$.
If $T_{y}=0$, then $T^{y_{1},\dots,y_{l-1}}(y)=0$. By Lemma 3.5, it follows
that

$\displaystyle\left|\\{y\in\mathbb{F}[t]_{Q}:T_{y}=0\\}\right|\leq\left|\\{y\in\mathbb{F}[t]_{Q}:T^{y_{1},\dots,y_{l-1}}(y)=0\\}\right|\leq
d_{l}\frac{|Q|}{\text{lpf}(Q)},$

unless $T^{y_{1},\dots,y_{l-1}}$ is the zero polynomial. So, it remains to
find $y_{1},\dots,y_{l-1}$ such that $T^{y_{1},\dots,y_{l-1}}$ is a nonzero
polynomial. Note that the coefficients of $T^{y_{1},\dots,y_{l-1}}$ are
polynomial expressions in $y_{1},\dots,y_{l-1}$ of degree at most $d_{i}$ in
the variable $y_{i}$. Since $T$ is not the zero polynomial, there is at least
one coefficient that is a nonzero polynomial
$C(y_{1},\dots,y_{l-1})\in(\mathbb{F}[t]_{Q})[y_{1},\dots,y_{l-1}]$. By the
induction hypothesis,

$\displaystyle\left|\left\\{(y_{1},\dots,y_{l-1})\in\mathbb{F}[t]_{Q}^{l-1}:C(y_{1},\dots,y_{l-1})=0\right\\}\right|\leq\left(\sum_{i=1}^{l-1}{d_{i}}\right)\frac{|Q|^{l-1}}{\text{lpf}(Q)}.$

Since $\text{lpf}(Q)>\sum_{i=1}^{l}{d_{i}}\geq\sum_{i=1}^{l-1}{d_{i}}$ by
assumption, it follows that $C(y_{1},\dots,y_{l-1})\neq 0$ for some
$(y_{1},\dots,y_{l-1})\in\mathbb{F}[t]_{Q}^{l-1}$. For this choice of
$y_{1},\dots,y_{l-1}$, the polynomial
$T^{y_{1},\dots,y_{l-1}}(y)\in(\mathbb{F}[t]_{Q})[y]$ is not the zero
polynomial, so we are done. ∎

Applying Lemma 3.5 to $\psi$, we get the bound

$\displaystyle\left|\left\\{(w_{1},\dots,w_{k-1})\in\mathbb{F}[t]_{Q}^{k-1}:\psi(w_{1},\dots,w_{k-1})=0\right\\}\right|\leq
p^{2m}(k-1)\frac{|Q|^{k-1}}{\text{lpf}(Q)}.$

Thus,

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e(P(y)/Q)}\right|^{2^{k-1}}$
$\displaystyle\leq\frac{\left|\left\\{(v_{1},\dots,v_{k-1})\in
K^{k-1}:\varphi(v_{1},\dots,v_{k-1})=0\right\\}\right|}{|K|^{k-1}}$
$\displaystyle=\frac{\left|\left\\{(w_{1},\dots,w_{k-1})\in\mathbb{F}[t]_{Q}^{k-1}:\psi(w_{1},\dots,w_{k-1})=0\right\\}\right|}{|Q|^{k-1}}$
$\displaystyle\leq\frac{p^{2m}(k-1)}{\text{lpf}(Q)}.$

∎

##  4\. Power saving bounds for the Furstenberg–Sárközy theorem in
characteristic $p$

We now prove Corollary 1.12, restated below:

See 1.12

###### Proof of Corollary 1.12.

Since $P$ is intersective, there exists $y\in\mathbb{F}[t]_{Q}$ with $P(y)=0$.
Hence, $-a_{0}=P(y)-a_{0}\in H_{Q}$. Therefore, applying Theorem 1.10 with
$f=\mathbbm{1}_{A}$, we have

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{x,y\in\mathbb{F}[t]_{Q}}{\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(x+P(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q},z\in
H_{Q}}{\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(x+z)}\right|\leq\frac{|A|}{|Q|}\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}\right)^{1/2^{k-1}}$

by the Cauchy–Schwarz inequality.

On the one hand, if $A$ contains no nontrivial patterns $\\{x,x+P(y)\\}$, then
by Lemma 3.5,

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{x,y\in\mathbb{F}[t]_{Q}}{\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(x+P(y))}=\frac{|A|}{|Q|}\frac{\left|\left\\{y\in\mathbb{F}[t]_{Q}:P(y)=0\right\\}\right|}{|Q|}\leq\frac{|A|}{|Q|}\frac{d}{\text{lpf}(Q)},$

as long as $\text{lpf}(Q)$ is large enough so that $P$ is not the zero
polynomial mod $Q$. On the other hand,

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}[t]_{Q},z\in
H_{Q}}{\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(x+z)}\geq\left(\frac{|A|}{|Q|}\right)^{2}$

by Lemma 3.3. Therefore,

$\displaystyle\left(\frac{|A|}{|Q|}\right)^{2}-\frac{|A|}{|Q|}\frac{d}{\text{lpf}(Q)}\leq
C\frac{|A|}{|Q|}\text{lpf}(Q)^{-1/2^{k-1}},$

where
$C=\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}(k-1)\right)^{1/2^{k-1}}$.
Multiplying both sides by $\frac{|Q|^{2}}{|A|}$, we get the desired bound

$\displaystyle|A|\leq
C|Q|\cdot\text{lpf}(Q)^{-1/2^{k-1}}+d\frac{|Q|}{\text{lpf}(Q)}\ll|Q|\cdot\text{lpf}(Q)^{-1/2^{k-1}}.$

∎

##  5\. Proof of equivalences

The goal of this section is to prove Theorem 1.17, restated here for
convenience:

See 1.17

First we prove that irrational equidistribution implies condition (iii).

######  Proposition 5.1.

Suppose $P(y)$ is good for irrational equidistribution, and let
$H\leq(\mathbb{F}[t],+)$ be the group generated by
$\\{P(y)-P(0):y\in\mathbb{F}[t]\\}$. Then there exists $C>0$ such that if
$Q(t)\in\mathbb{F}[t]^{+}$ satisfies $\text{lpf}(Q)\geq C$, then
$H+Q\mathbb{F}[t]=\mathbb{F}[t]$. That is, (iii) holds.

###### Proof.

We prove the contrapositive. Suppose (iii) fails. Then there is a sequence
$(Q_{n})_{n\in\mathbb{N}}$ in $\mathbb{F}[t]^{+}$ such that
$\text{lpf}(Q_{n})\to\infty$ and $H+Q_{n}\mathbb{F}[t]\neq\mathbb{F}[t]$ for
$n\in\mathbb{N}$. Equivalently,

$\displaystyle
H^{\perp}\cap(Q_{n}\mathbb{F}[t])^{\perp}=(H+Q_{n}\mathbb{F}[t])^{\perp}\neq\\{0\\}.$

Since $(Q_{n}\mathbb{F}[t])^{\perp}\cong\widehat{\mathbb{F}[t]}_{Q_{n}}$, it
follows that $\frac{s_{n}}{Q_{n}}\in H^{\perp}$ for some $s_{n}\not\equiv
0\pmod{Q_{n}}$. If $\frac{s_{n}}{Q_{n}}=\frac{s^{\prime}_{n}}{Q^{\prime}_{n}}$
in reduced terms (i.e., $s^{\prime}_{n}$ and $Q^{\prime}_{n}$ are coprime),
then $\text{lpf}(Q^{\prime}_{n})\geq\text{lpf}(Q_{n})$, so we may assume
without loss of generality that $s_{n}$ and $Q_{n}$ are coprime. The sequence
$\left(\frac{s_{n}}{Q_{n}}\right)_{n\in\mathbb{N}}$ then consists of distinct
elements, so $H^{\perp}$ is infinite. Every infinite compact group is
uncountable, and there are only countable many rational points, so $H^{\perp}$
must contain an irrational element. That is, for some irrational $\alpha$,
$e(P(y)\alpha)=e(P(0)\alpha)$ for every $y\in\mathbb{F}[t]$. Hence,
$\left(P(n)\alpha\right)_{n\in\mathbb{N}}$ is not well distributed, so $P(y)$
is not good for irrational equidistribution. ∎

We will now prove the equivalences in Theorem 1.17 by showing the implications
illustrated in the following diagram:

${(i)}$${(ii)}$${(iii)}$${(v)}$${(iv)}$

Condition (ii) is a quantitative refinement of condition (i), so we
immediately have the implication (ii)$\implies$(i). By Theorem 1.10, we have
the additional implications (i)$\implies$(iii)$\implies$(ii).

Condition (v) follows from (ii) by a straightforward application of the
Cauchy–Schwarz inequality.

######  Proposition 5.2.

(v)$\implies$(iv).

###### Proof.

Let $\delta>0$. Let $C_{1}$, $C_{2}$, and $\gamma$ be as in (v). Let
$Q(t)\in\mathbb{F}[t]^{+}$ with $\text{lpf}(Q)\geq C_{1}$. Let
$A,B\subseteq\mathbb{F}[t]_{Q}$ with $|A||B|\geq\delta|Q|^{2}$. Then by (v),

$\displaystyle\left|\left\\{(x,y)\in\mathbb{F}[t]_{Q}^{2}:x\in A,x+P(y)\in
B\right\\}\right|$
$\displaystyle\geq|A||B|-C_{2}|A|^{1/2}|B|^{1/2}|Q|\cdot\text{lpf}(Q)^{-\gamma}$
$\displaystyle=|A|^{1/2}|B|^{1/2}\left(|A|^{1/2}|B|^{1/2}-C_{2}|Q|\cdot\text{lpf}(Q)^{-\gamma}\right)$
$\displaystyle\geq\delta^{1/2}\left(\delta^{1/2}-C_{2}\cdot\text{lpf}(Q)^{-\gamma}\right)|Q|^{2}.$

Thus, if

$\displaystyle\text{lpf}(Q)\geq
N(\delta)=\max{\left\\{C_{1},\left(\frac{C_{2}}{\delta^{1/2}}\right)^{1/\gamma}\right\\}},$

then we can find $x,y\in\mathbb{F}[t]_{Q}$ with $x\in A$ and $x+P(y)\in B$. ∎

We now prove the final implication to complete the proof of Theorem 1.17:

######  Proposition 5.3.

(iv)$\implies$(iii).

###### Proof.

We prove the contrapositive. Suppose (iii) fails. Then there is a sequence
$(Q_{n})_{n\in\mathbb{N}}$ in $\mathbb{F}[t]^{+}$ with
$\text{lpf}(Q_{n})\to\infty$ such that
$\left\\{P(y)-P(0):y\in\mathbb{F}[t]_{Q_{n}}\right\\}\subseteq
H_{Q_{n}}\lneq\mathbb{F}[t]_{Q_{n}}$ for every $n\in\mathbb{N}$. The subgroup
$H_{Q_{n}}$ has index $p^{k_{n}}$ for some $k_{n}\in\mathbb{N}$. Let $A_{n}$
be a union of $\left\lfloor\frac{p^{k_{n}}}{2}\right\rfloor$ cosets of
$H_{Q_{n}}$, and let
$B_{n}=\mathbb{F}[t]_{Q_{n}}\setminus\left(A_{n}+P(0)\right)$. For any $x\in
A_{n}$ and $y\in\mathbb{F}[t]_{Q_{n}}$, we have

$\displaystyle x+P(y)=x+P(0)+(P(y)-P(0))\in A_{n}+P(0)+H_{Q_{n}}=A_{n}+P(0).$

That is, if $x\in A_{n}$ and $y\in\mathbb{F}[t]_{Q_{n}}$, then $x+P(y)\notin
B_{n}$. Moreover,

$\displaystyle\frac{|A_{n}|}{|Q_{n}|}=\frac{\left\lfloor\frac{p^{k_{n}}}{2}\right\rfloor}{p^{k_{n}}}\geq\frac{1}{3}\qquad\text{and}\qquad\frac{|B_{n}|}{|Q_{n}|}\geq\frac{1}{2}.$

Therefore, property (iv) fails for $\delta=\frac{1}{6}$. ∎

Now we proceed to prove the remaining equivalences in Theorem 1.18. As a first
step, we have the following characterization of irrational equidistribution
for additive polynomials:

######  Proposition 5.4.

Let $\eta(y)\in(\mathbb{F}[t])[y]$ be an additive polynomial. The following
are equivalent:

  1. (i)

$\eta(y)$ is good for irrational equidistribution;

  2. (ii)

there exists $C>0$ such that if $Q(t)\in\mathbb{F}[t]^{+}$ and
$\text{lpf}(Q)\geq C$, then $\eta(\mathbb{F}[t]_{Q})=\mathbb{F}[t]_{Q}$;

  3. (iii)

$\eta(y)=ay$ for some $a\in\mathbb{F}[t]\setminus\\{0\\}$.

###### Proof.

(i)$\implies$(ii). See Proposition 5.1.

(ii)$\implies$(iii). We prove the contrapositive. Suppose
$\eta(y)=\sum_{i=0}^{k}{a_{i}y^{p^{i}}}$ with $a_{k}\neq 0$, $k\geq 1$. We
consider two cases separately.

Case 1: $a_{0}=0$.

In this case, we may write $\eta(y)=\eta^{\prime}(y)^{p}$, where
$\eta^{\prime}(y)=\sum_{i=1}^{k}{a_{i}y^{p^{i-1}}}$. Hence, for any
$Q(t)\in\mathbb{F}[t]^{+}$,
$\eta(\mathbb{F}[t]_{Q})\subseteq\\{y^{p}:y\in\mathbb{F}[t]_{Q}\\}$. If
$Q=Q_{0}^{2}$, then $Q_{0}\not\equiv 0\pmod{Q}$, while $Q_{0}^{p}\equiv
0\pmod{Q}$. Therefore, the homomorphism $y\mapsto y^{p}$ has a nontrivial
kernel mod $Q$, so $\\{y^{p}:y\in\mathbb{F}[t]_{Q}\\}$ is a proper subgroup of
$\mathbb{F}[t]_{Q}$. Taking $Q_{0}$ to be an arbitrarily large irreducible
element of $\mathbb{F}[t]^{+}$, this shows that (ii) does not hold.

Case 2: $a_{0}\neq 0$.

Write $\eta(y)=yg(y)$ with $g(y)=a_{0}+\sum_{i=1}^{k}{a_{i}y^{p^{i}-1}}$.
Since $g(y)$ is a nonconstant polynomial, the set

$\displaystyle R=\left\\{Q(t)\in\mathbb{F}[t]^{+}:Q~{}\text{is irreducible
and}~{}g~{}\text{has a root mod}~{}Q\right\\}$

is infinite. Indeed, for any finite collection of irreducibles
$Q_{1},\dots,Q_{r}\in\mathbb{F}[t]^{+}$, consider

(5.1) $\displaystyle g\left(a_{0}Q_{1}\dots
Q_{r}y\right)=a_{0}\left(1+Q_{1}\dots
Q_{r}y\sum_{i=1}^{k}{a_{i}\left(a_{0}Q_{1}\dots,Q_{r}y\right)^{p^{i}-2}}\right).$

Since $g$ is a nonzero polynomial, there exists $y_{0}\in\mathbb{F}[t]$ such
that $g\left(a_{0}Q_{1}\dots Q_{r}y_{0}\right)\neq 0$. From the expression on
the right hand side of (5.1), we have $a_{0}\mid g\left(a_{0}Q_{1}\dots
Q_{r}y_{0}\right)$, and $\frac{g\left(a_{0}Q_{1}\dots
Q_{r}y_{0}\right)}{a_{0}}\equiv 1\pmod{Q_{i}}$ for each $i\in\\{1,\dots,r\\}$.
Therefore, there is some irreducible
$Q\in\mathbb{F}[t]^{+}\setminus\\{Q_{1},\dots,Q_{r}\\}$ such that $Q\mid
g\left(a_{0}Q_{1}\dots Q_{r}y_{0}\right)$. Hence, $R$ is infinite as claimed.

Suppose $Q\in R$ and $Q\nmid a_{0}$. Let $y\in\mathbb{F}[t]_{Q}$ such that
$g(y)=0$. Then $\eta(y)=yg(y)=0$, but $y\neq 0$, since $y\mid a_{0}$. Hence,
$\eta$ has a nontrivial kernel mod $Q$ for infinitely many irreducibles $Q$,
which contradicts condition (ii).

(iii)$\implies$(i). See [BL16, Theorem 0.1]. ∎

The following lemma is the key tool to reduce equidistributional properties of
polynomials to the additive case with which we have just dealt.

######  Lemma 5.5.

Let $\eta_{1},\eta_{2}\in\mathbb{F}_{p}[y]$ be additive polynomials, and let
$H_{i}=\eta_{i}(\mathbb{F}[t])$ for $i=1,2$. There exists an additive
polynomial $\eta\in\mathbb{F}_{p}[y]$ such that
$\eta(\mathbb{F}[t])=H_{1}+H_{2}$. Moreover,
$\eta=\eta_{1}\circ\zeta_{1}+\eta_{2}\circ\zeta_{2}$, where
$\zeta_{1},\zeta_{2}\in\mathbb{F}_{p}[y]$ are additive polynomials.

###### Proof.

If $\eta_{i}=0$ for some $i$, then take $\eta=\eta_{j}$ with $j\neq i$.

Suppose now that $\eta_{1}$ and $\eta_{2}$ are both nonzero. Let
$H=H_{1}+H_{2}$. Write $\eta_{1}(y)=\sum_{i=0}^{k}{a_{i}y^{p^{i}}}$ and
$\eta_{2}(y)=\sum_{j=0}^{l}{b_{j}y^{p^{j}}}$. Without loss of generality,
$k\geq l$. Define

(5.2)
$\displaystyle\eta^{\prime}_{1}(y)=b_{l}\eta_{1}(y)-a_{k}\eta_{2}\left(y^{p^{k-l}}\right)=\eta_{1}(b_{l}y)+\eta_{2}\left(-a_{k}y^{p^{k-l}}\right),$

and let $H^{\prime}_{1}=\eta^{\prime}_{1}(\mathbb{F}[t])$. Then
$\deg{\eta^{\prime}_{1}}<\deg{\eta_{1}}$.

Claim: $H^{\prime}_{1}+H_{2}=H_{1}+H_{2}$.

For any $y\in\mathbb{F}[t]$, (5.2) expressed $\eta^{\prime}_{1}(y)$ as a sum
of an element of $H_{1}$ and an element of $H_{2}$. Hence,
$H^{\prime}_{1}\subseteq H_{1}+H_{2}$. Rearranging (5.2), we have

$\displaystyle\eta_{1}(y)=b_{l}^{-1}\eta^{\prime}_{1}(y)+b_{l}^{-1}a_{k}\eta_{2}\left(y^{p^{k-l}}\right).$

Thus, $H_{1}\subseteq H^{\prime}_{1}+H_{2}$. This proves the claim.

We have shown that, given any nonzero additive polynomials
$\eta_{1},\eta_{2}\in\mathbb{F}_{p}[y]$, we may find
$\eta^{\prime}_{1},\eta^{\prime}_{2}\in\mathbb{F}_{p}[y]$ with
$\eta^{\prime}_{1}(\mathbb{F}[t])+\eta^{\prime}_{2}(\mathbb{F}[t])=\eta_{1}(\mathbb{F}[t])+\eta_{2}(\mathbb{F}[t])$
such that
$\deg{\eta^{\prime}_{1}}+\deg{\eta^{\prime}_{2}}<\deg{\eta_{1}}+\deg{\eta_{2}}$,
and $\eta^{\prime}_{1}$ and $\eta^{\prime}_{2}$ are of the appropriate form.
Repeating this process finitely many times, we eventually reduce to the
situation that one of the additively polynomials is zero. We then take $\eta$
to be the remaining nonzero polynomial. ∎

The argument in the proof of Lemma 5.5 provides an algorithm for obtaining
$\eta$ that bears a strong resemblance with the Euclidean algorithm. We work
through a few simple examples to see more concretely how the algorithm works.

######  Example 5.6.

(1) $\eta_{1}(y)=y^{p^{2}}-y$, $\eta_{2}(y)=y^{p^{3}}+y^{p}$. The polynomial
$\eta_{2}$ has larger degree, so we shift the exponents of $\eta_{1}$ to match
the degree of $\eta_{2}$ and subtract:

$\displaystyle\eta^{\prime}_{2}(y)=\eta_{2}(y)-\eta_{1}(y^{p})=2y^{p}.$

If $p=2$, then $\eta^{\prime}_{2}(y)=0$, so we stop, and the resulting
polynomial $\eta$ is simply $\eta_{1}$. (Note that when $p=2$, $\eta_{1}$ may
be rewritten as $\eta_{1}(y)=y^{p^{2}}+y$, and then it is clear that
$\eta_{2}(y)=\eta_{1}(y^{p})$, so the range of $\eta_{2}$ is manifestly a
subset of the range of $\eta_{1}$.) Suppose $p>2$. Then
$\deg{\eta_{1}}>\deg{\eta^{\prime}_{2}}$, so we shift the exponents of
$\eta^{\prime}_{2}$ and subtract:

$\displaystyle\eta^{\prime}_{1}(y)=2\eta_{1}(y)-\eta^{\prime}_{2}(y^{p})=-2y.$

Since $p>2$, the element $-2\in\mathbb{F}_{p}$ is invertible, so the image of
$\eta^{\prime}_{1}$ is all of $\mathbb{F}[t]$, and we are done:
$\eta(y)=\eta^{\prime}_{1}(y)=-2y$. (One can check that applying one more step
of the algorithm would result in $\eta^{\prime\prime}_{2}=0$, indicating that
the process has terminated.)

(2) $\eta_{1}(y)=y^{p^{3}}+y^{p^{2}}+y^{p}$, $\eta_{2}(y)=y^{p^{2}}$. First,
shifting $\eta_{2}$ and subtracting, we have

$\displaystyle\eta^{\prime}_{1}(y)=\eta_{1}(y)-\eta_{2}(y^{p})=y^{p^{2}}+y^{p}.$

Next, subtracting $\eta_{2}$ without any shifting gives

$\displaystyle\eta^{\prime\prime}_{1}(y)=\eta^{\prime}_{1}(y)-\eta_{2}(y)=y^{p}.$

Shifting $\eta^{\prime\prime}_{1}$ and subtracting from $\eta_{2}$ produces
$\eta^{\prime}_{2}=0$, so we are done and
$\eta(y)=\eta^{\prime\prime}_{1}(y)=y^{p}$.

The following proposition completes the proof of Theorem 1.18:

######  Proposition 5.7.

Let $P(y)\in\mathbb{F}_{p}[y]$ and write
$P(y)=a_{0}+\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$ with
$\eta_{1},\dots,\eta_{n}\in\mathbb{F}_{p}[y]$ additive polynomials and
$r_{1},\dots,r_{n}\in\mathbb{N}$ distinct positive integers not divisible by
$p$. The following are equivalent:

  1. (i)

$P(y)$ is good for irrational equidistribution;

  2. (ii)

there exists $C>0$ such that if $Q(t)\in\mathbb{F}[t]^{+}$ and
$\text{lpf}(Q)\geq C$, then $H+Q\mathbb{F}[t]=\mathbb{F}[t]$, where
$H\leq(\mathbb{F}[t],+)$ is the group generated by
$\\{P(y)-P(0):y\in\mathbb{F}[t]\\}$;

  3. (iii)

there exist additive polynomials
$\zeta_{1},\dots,\zeta_{n}\in\mathbb{F}_{p}[y]$ and
$a\in\mathbb{F}_{p}^{\times}$ such that

$\displaystyle\sum_{i=1}^{n}{(\eta_{i}\circ\zeta_{i})(y)}=ay.$

###### Proof.

(i)$\implies$(ii). See Theorem 1.17.

(ii)$\implies$(iii). For $Q(t)\in\mathbb{F}[t]^{+}$ and $i\in\\{1,\dots,n\\}$,
let $H_{i,Q}=\eta_{i}(\mathbb{F}[t]_{Q})$, By (ii), we have that
$H_{Q}=\mathbb{F}[t]_{Q}$ whenever $\text{lpf}(Q)\geq C$. By Lemma 5.5, there
is an additive polynomial of the form

$\displaystyle\eta=\sum_{i=1}^{n}{\eta_{i}\circ\zeta_{i}}$

such that $H_{Q}=\eta(\mathbb{F}[t]_{Q})$ for every
$Q(t)\in\mathbb{F}[t]^{+}$. In particular,
$\eta(\mathbb{F}[t]_{Q})=\mathbb{F}[t]_{Q}$ for all $Q$ with
$\text{lpf}(Q)\geq C$. Hence, by Proposition 5.4, $\eta(y)=ay$. That is, (iii)
holds.

(iii)$\implies$(i). Let $\alpha\in\mathbb{F}((t^{-1}))\setminus\mathbb{F}(t)$.
Let

$\displaystyle\mathcal{F}_{i}=\overline{\eta_{i}(\mathbb{F}[t])\alpha}$

By Theorem 1.9, $\left(P(y)\alpha\right)_{y\in\mathbb{F}[t]}$ is well
distributed if and only if $\mathcal{F}=\sum_{i=1}^{n}{\mathcal{F}_{i}}$ is
the full “torus” $\mathbb{F}((t^{-1}))/\mathbb{F}[t]$. By (iii),

$\displaystyle\mathcal{F}\ni\sum_{i=1}^{n}{\eta_{i}\left(\zeta_{i}(y)\right)}\alpha=ay\alpha$

for every $y\in\mathbb{F}[t]$. But $(ay\alpha)_{y\in\mathbb{F}[t]}$ is well-
distributed (in particular, it is dense) mod $\mathbb{F}[t]$ (see [BL16,
Theorem 0.1]), so $\mathcal{F}=\mathbb{F}((t^{-1}))/\mathbb{F}[t]$. Thus,
$P(y)$ is good for irrational equidistribution. ∎

##  6\. Partition regularity of polynomial equations over finite fields

In this section, we deduce additional applications of Theorem 1.10 to
partition regularity of polynomial equations over finite fields. As a first
step, we observe the following criterion for the existence of solutions to
polynomial equations of the form $P_{1}(x)+P_{2}(y)+P_{3}(z)=c$ over finite
fields:

######  Proposition 6.1.

Let $P_{1},P_{2},P_{3}\in(\mathbb{F}[t])[x]$ be nonconstant polynomials with
$P_{i}(0)=0$. Let $d_{i}=\deg{P_{i}}$ and $k_{i}=\text{d-}\deg{P_{i}}$. Let
$H_{q,i}=\left\langle
P_{i}(x):x\in\mathbb{F}_{q}\right\rangle\leq(\mathbb{F}_{q},+)$, and let
$H_{q}=H_{q,1}+H_{q,2}+H_{q,3}$. Then for any $q$ and any $c\in H_{q}$,

$\displaystyle\left|\left\\{(x,y,z)\in\mathbb{F}_{q}^{3}:P_{1}(x)+P_{2}(y)+P_{3}(z)=c\right\\}\right|=\frac{q^{3}}{|H_{q}|}+O_{q\to\infty;d_{1},d_{2},d_{3},k_{1},k_{2},k_{3}}\left(q^{2-\gamma}\right),$

where $\gamma=\min{\left\\{\frac{1}{2^{k_{i}-1}}:i\in\\{1,2,3\\}\right\\}}>0$.
In particular, if $q$ is sufficiently large, then

$\displaystyle
P_{1}(\mathbb{F}_{q})+P_{2}(\mathbb{F}_{q})+P_{3}(\mathbb{F}_{q})=H_{q}.$

######  Remark 6.2.

The main term, $\frac{q^{3}}{|H_{q}|}$, expresses an asymptotic
equidistribution property. There are $q^{3}$ choices of
$(x,y,z)\in\mathbb{F}_{q}^{3}$ and $|H_{q}|$ possible values $c\in H_{q}$ that
are free from obvious algebraic obstructions to solvability of
$P_{1}(x)+P_{2}(y)+P_{3}(z)=c$. Proposition 6.1 states that, asymptotically,
the polynomial $P_{1}(x)+P_{2}(y)+P_{3}(z)$ takes on each such value of $c$
with roughly equal frequency.

###### Proof.

For any prime power $q$, any $c\in\mathbb{F}_{q}$, and any functions
$f_{1},\dots,f_{r}:\mathbb{F}_{q}\to\mathbb{F}_{q}$, let
$N(q,c;f_{1},\dots,f_{r})$ denote the number of solutions
$(x_{1},\dots,x_{r})\in\mathbb{F}_{q}^{r}$ to the equation
$\sum_{i=1}^{r}{f_{i}(x_{i})}=c$. Our goal is to show

$\displaystyle
N(q,c;P_{1},P_{2},P_{3})=\frac{q^{3}}{|H_{q}|}+O_{q\to\infty}\left(q^{2-\gamma}\right)$

for $c\in H_{q}$.

For each $i\in\\{1,2,3\\}$, let $\eta_{i}$ be an additive polynomial with
$\eta_{i}(\mathbb{F}_{q})=H_{q,i}$.

Claim:
$N(q,c;P_{1},P_{2},P_{3})=N(q,c;P_{1},P_{2},\eta_{3})+O_{q\to\infty;d_{1},d_{2},d_{3},k_{3}}\left(q^{2-1/2^{k_{3}-1}}\right)$.

Let $f_{i}(x)=N(q,x;P_{i})$. Then

$\displaystyle
N(q,c;P_{1},P_{2},P_{3})=\sum_{x,y\in\mathbb{F}_{q}}{f_{1}(x)f_{2}(c-x-
P_{3}(y))}=q^{2}\operatorname*{\text{\Large$\mathbb{E}$}}_{x,y\in\mathbb{F}_{q}}{f_{1}(x)f_{2}(c-x-
P_{3}(y))}.$

Similarly,

$\displaystyle
N(q,c;P_{1},P_{2},\eta_{3})=q^{2}\operatorname*{\text{\Large$\mathbb{E}$}}_{x,y\in\mathbb{F}_{q}}{f_{1}(x)f_{2}(c-x-\eta_{3}(y))}=q^{2}\operatorname*{\text{\Large$\mathbb{E}$}}_{x\in\mathbb{F}_{q}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{q,3}}{f_{1}(x)f_{2}(c-x-z)}}.$

Hence, by the Cauchy–Schwarz inequality,

$\displaystyle\left|N(q,c;P_{1},P_{2},P_{3})-N(q,c;P_{1},P_{2},\eta_{3})\right|\leq
q^{2}\left\|f_{1}\right\|_{L^{2}(\mathbb{F}_{q})}\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}_{q}}{f_{2}(c-x-
P_{3}(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{q,3}}{f_{2}(c-x-z)}\right\|_{L^{2}(\mathbb{F}_{q})}.$

Now, by Theorem 1.10,

$\displaystyle\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}_{q}}{f_{2}(c-x-
P_{3}(y))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{q,3}}{f_{2}(c-x-z)}\right\|_{L^{2}(\mathbb{F}_{q})}\ll_{d_{3},k_{3}}q^{-1/2^{k_{3}-1}}\left\|f_{2}\right\|_{L^{2}(\mathbb{F}_{q})}.$

Finally, for each $x\in\mathbb{F}_{q}$, the polynomial equation $P_{i}(u)=x$
has at most $d_{i}$ solutions $u\in\mathbb{F}_{q}$, so
$\left\|f_{i}\right\|_{L^{2}(\mathbb{F}_{q})}\leq\left\|f_{i}\right\|_{L^{\infty}(\mathbb{F}_{q})}\leq
d_{i}$. Putting everything together,

$\displaystyle
N(q,c;P_{1},P_{2},P_{3})=N(q,c;P_{1},P_{2},\eta_{3})+O_{q\to\infty;d_{1},d_{2},d_{3},k_{3}}\left(q^{2-1/2^{k_{3}-1}}\right)$

as claimed.

Applying the claim also to $P_{1}$ and $P_{2}$, we obtain the estimate

$\displaystyle
N(q,c;P_{1},P_{2},P_{3})=N(q,c;\eta_{1},\eta_{2},\eta_{3})+O_{q\to\infty;d_{1},d_{2},d_{3},k_{1},k_{2},k_{3}}\left(q^{2-\gamma}\right).$

Let $\eta:\mathbb{F}_{q}^{3}\to\mathbb{F}_{q}$,
$\eta(x,y,z)=\eta_{1}(x)+\eta_{2}(y)+\eta_{3}(z)$. Then $\eta$ is a group
homomorphism with image $H_{q}$. Therefore,
$N(q,c;\eta_{1},\eta_{2},\eta_{3})=\left|\eta^{-1}(\\{c\\})\right|$ is
constant in $c\in H_{q}$, so

$\displaystyle
N(q,c;\eta_{1},\eta_{2},\eta_{3})=\frac{\left|\mathbb{F}_{q}^{3}\right|}{|H_{q}|}=\frac{q^{3}}{|H_{q}|}.$

∎

Problems in the vein of Proposition 6.1 counting solutions to polynomial
equations over finite fields are well-studied. For instance, if the equation
$P_{1}(x)+P_{2}(y)+P_{3}(z)=c$ defines a geometrically irreducible variety,
then a theorem of Lang and Weil [LW54] states that the number of solutions is
equal to $q^{2}+O_{q\to\infty}(q^{3/2})$. The class of polynomials handled by
Proposition 6.1 is very restricted, and the error term in Proposition 6.1 is
in general much weaker that what can obtained with the use of algebraic
geometry. However, the elementary method of proof allows us to avoid any
irreducibility assumption and is more flexible for combinatorial enhancements,
such as the following Ramsey-theoretic result, restated from the introduction:

See 1.19

Our proof of Theorem 1.19 combines Theorem 1.10 with tools from the theory of
Loeb measures on ultraproduct spaces. We will need the following
generalization of Theorem 1.10, which in fact comes as an easy consequence of
the key estimate on character sums in Proposition 3.2:

######  Proposition 6.3.

Let $P(y)\in(\mathbb{F}[t])[y]$ be a nonconstant polynomial of degree $d$ and
derivational degree $k$. Write
$P(y)=a_{0}+\sum_{i=1}^{n}{\eta_{i}(y^{r_{i}})}$. Let
$H_{i}=\eta_{i}\left(\mathbb{F}[t]\right)$ and $H=\sum_{i=1}^{n}{H_{i}}$. Then
for any $Q(t)\in\mathbb{F}[t]^{+}$, any $m\in\mathbb{N}$, and any
$f:\mathbb{F}[t]_{Q}^{m}\to\mathbb{C}$,

$\displaystyle\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x_{1}+P(y),\dots,x_{m}+P(y)))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{Q}}{f(x_{1}+a_{0}+z,\dots,x_{m}+a_{0}+z)}\right\|_{L^{2}(\mathbb{F}[t]_{Q}^{m})}$
$\displaystyle\leq\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}\right)^{1/2^{k-1}}\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q}^{m})},$

where $H_{Q}=\left\\{z\pmod{Q}:z\in H\right\\}$.

###### Proof.

Expand $f$ as a Fourier series:

$\displaystyle
f(x_{1},\dots,x_{m})=\sum_{s_{1},\dots,s_{m}\in\mathbb{F}[t]_{Q}}{\widehat{f}(s_{1},\dots,s_{m})e\left(\sum_{j=1}^{m}{s_{j}x_{j}}/Q\right)}.$

Then for $\mathbf{x}=(x_{1},\dots,x_{m})\in\mathbb{F}[t]_{Q}^{m}$, we have

$\displaystyle F(\mathbf{x})$
$\displaystyle:=\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{f(x_{1}+P(y),\dots,x_{m}+P(y)))}-\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in
H_{Q}}{f(x_{1}+a_{0}+z,\dots,x_{m}+a_{0}+z)}$
$\displaystyle=\sum_{\mathbf{s}\in\mathbb{F}[t]_{Q}^{m}}{\widehat{f}(\mathbf{s})e(\mathbf{s}\cdot\mathbf{x}/Q)\left(\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e\left(\sum_{j=1}^{m}{s_{j}}P(y)/Q\right)}-e\left(\sum_{j=1}^{m}{s_{j}}a_{0}\right)\mathbbm{1}_{H_{Q}^{\perp}}\left(\sum_{j=1}^{m}{s_{j}}\right)\right)}.$

For each $\mathbf{s}\in\mathbb{F}[t]_{Q}^{m}$, Proposition 3.2 gives the bound

$\displaystyle\left|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}[t]_{Q}}{e\left(\sum_{j=1}^{m}{s_{j}}P(y)/Q\right)}-e\left(\sum_{j=1}^{m}{s_{j}}a_{0}\right)\mathbbm{1}_{H_{Q}^{\perp}}\left(\sum_{j=1}^{m}{s_{j}}\right)\right|\leq\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}\right)^{1/2^{k-1}}.$

Therefore, by Parseval’s identity,

$\displaystyle\left\|F\right\|_{L^{2}(\mathbb{F}[t]_{Q}^{m})}^{2}\leq\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}\right)^{1/2^{k-2}}\sum_{\mathbf{s}\in\mathbb{F}[t]_{Q}^{m}}{\left|\widehat{f}(\mathbf{s})\right|^{2}}=\left(p^{2\left\lfloor\log_{p}{d}\right\rfloor}\frac{k-1}{\text{lpf}(Q)}\right)^{1/2^{k-2}}\left\|f\right\|_{L^{2}(\mathbb{F}[t]_{Q}^{m})}^{2}.$

∎

The relevant constructions for employing measure theory on ultraproducts are
summarized as follows:

######  Definition 6.4.

  * •

An _ultrafilter_ on $\mathbb{N}$ is a collection
$\mathcal{U}\subseteq\mathcal{P}(\mathbb{N})$ of nonempty subsets of
$\mathbb{N}$ such that:

    * –

if $A,B\in\mathcal{U}$, then $A\cap B\in\mathcal{U}$;

    * –

for any $A\subseteq\mathbb{N}$, either $A\in\mathcal{U}$ or
$\mathbb{N}\setminus A\in\mathcal{U}$.

The ultrafilter $\mathcal{U}$ is _principal_ if
$\mathcal{U}=\\{A\subseteq\mathbb{N}:n\in A\\}$ for some $n\in\mathbb{N}$ and
_non-principal_ otherwise. The space of ultrafilters is denoted
$\beta\mathbb{N}$.

  * •

Given $\mathcal{U}\in\beta\mathbb{N}$ and a family of sets
$(X_{n})_{n\in\mathbb{N}}$, the _ultraproduct_ is the set

$\displaystyle\prod_{n\to\mathcal{U}}{X_{n}}=\left(\prod_{n\in\mathbb{N}}{X_{n}}\right)/\equiv_{\mathcal{U}},$

where $\equiv_{\mathcal{U}}$ is the equivalence relation defined by
$(x_{n})_{n\in\mathbb{N}}\equiv_{\mathcal{U}}(y_{n})_{n\in\mathbb{N}}$ if and
only if $\\{n\in\mathbb{N}:x_{n}=y_{n}\\}\in\mathcal{U}$.

  * •

Given $\mathcal{U}\in\beta\mathbb{N}$ and a sequence
$(x_{n})_{n\in\mathbb{N}}$ taking values in a compact Hausdorff space $X$, the
_limit of $(x_{n})_{n\in\mathbb{N}}$ along $\mathcal{U}$_ is defined to be the
unique point666Such a point $x$ exists by compactness and is unique by the
Hausdorff property. $x\in X$ such that for any neighborhood $U$ of $x$, one
has $\\{n\in\mathbb{N}:x_{n}\in U\\}\in\mathcal{U}$. The limit of
$(x_{n})_{n\in\mathbb{N}}$ along $\mathcal{U}$ is denoted by
$\lim_{n\to\mathcal{U}}{x_{n}}$.

  * •

Let $\mathcal{U}\in\beta\mathbb{N}$, and let
$(X_{n},\mathcal{X}_{n},\mu_{n})_{n\in\mathbb{N}}$ be a family of probability
spaces. Let $X=\prod_{n\to\mathcal{U}}{X_{n}}$.

    * –

An _internal set_ is a set of the form $\prod_{n\to\mathcal{U}}{A_{n}}$ with
$A_{n}\in\mathcal{X}_{n}$.

    * –

The _Loeb $\sigma$-algebra_ $\mathcal{X}$ is the $\sigma$-algebra on
$\prod_{n\to\mathcal{U}}{X_{n}}$ generated by the algebra of internal sets.

    * –

The _Loeb measure_ $\mu$ is the unique probability measure on $\mathcal{X}$
with the property

$\displaystyle\mu(A)=\lim_{n\to\mathcal{U}}{\mu_{n}(A_{n})}$

for any internal set $A=\prod_{n\to\mathcal{U}}{A_{n}}$.

The main property of the Loeb measure that we will use is the following
version of Fubini’s theorem:

######  Proposition 6.5 (cf. [K77], Theorem 1.12).

Let $(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$ be sequences of
finite sets. Let $\mathcal{U}$ be a non-principal ultrafilter. Let
$X=\prod_{n\to\mathcal{U}}{X_{n}}$ and $Y=\prod_{n\to\mathcal{U}}{Y_{n}}$. Let
$f:X\times Y\to\mathbb{C}$ be a bounded Loeb-measurable function. Then

  1. (1)

for any $x\in X$, the function $y\mapsto f(x,y)$ is Loeb-measurable on $Y$;

  2. (2)

the function $x\mapsto\int_{Y}{f(x,y)~{}d\mu_{Y}(y)}$ is Loeb-measurable on
$X$; and

  3. (3)

$\displaystyle\int_{X\times Y}{f~{}d\mu_{X\times
Y}}=\int_{X}{\left(\int_{Y}{f(x,y)~{}d\mu_{Y}(y)}\right)~{}d\mu_{X}(x)}.$

######  Remark 6.6.

Proposition 6.5 does not follow from standard version of Fubini’s theorem. The
subtlety lies in the structure of the Loeb $\sigma$-algebra on the product
space $X\times Y$: there are internal subsets of $X\times Y$ that cannot be
approximated by Boolean combinations of Cartesian products of internal subsets
of $X$ and $Y$ (on the finitary level, this corresponds to approximating
subsets of $X_{n}\times Y_{n}$ by products of boundedly many subsets of
$X_{n}$ and $Y_{n}$). Therefore, the function $f$ need not be measurable with
respect to the product of the Loeb $\sigma$-algebras on $X$ and $Y$.
Nevertheless, Proposition 6.5 shows that $\mu_{X\times Y}$ shares important
features with the product measure $\mu_{X}\times\mu_{Y}$.

###### Proof of Theorem 1.19.

Let $r\in\mathbb{N}$. Suppose for contradiction that there are $r$-colorings
$\mathbb{F}_{p^{k_{n}}}=\bigcup_{i=1}^{r}{C_{n,i}}$ with $k_{n}\to\infty$ such
that $|M_{n}|=o_{n\to\infty}\left(\left(p^{k_{n}}\right)^{2}\right)$, where

$\displaystyle
M_{n}=\left\\{(x,y,z)\in\mathbb{F}_{p^{k_{n}}}^{3}:P(x)-P(y)=Q(z)~{}\text{and}~{}\\{x,y,z\\}\subseteq
C_{n,i}~{}\text{for some}~{}i\in\\{1,\dots,r\\}\right\\}$

is the collection of monochromatic solutions to the equation $P(x)-P(y)=Q(z)$.

Now we define a limit object associated with this sequence of colorings. Fix a
non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$. Let
$\mathbb{F}_{\infty}$ be the pseudo-finite field
$\mathbb{F}_{\infty}=\prod_{n\to\mathcal{U}}{\mathbb{F}_{p^{k_{n}}}}$, let
$C_{i}=\prod_{n\to\mathcal{U}}{C_{n,i}}\subseteq\mathbb{F}_{\infty}$, and let
$M=\prod_{n\to\mathcal{U}}{M_{n}}$. Denote by $\mu$ the Loeb measure on
$\mathbb{F}_{\infty}$ obtained by equipping $\mathbb{F}_{p^{k_{n}}}$ with the
normalized counting measure. For any $s\in\mathbb{N}$, we denote the Loeb
measure on $\mathbb{F}_{\infty}^{s}$ by $\mu^{s}$ (not be confused with the
product measure $\mu\times\dots\times\mu$ on $\mathbb{F}_{\infty}^{s}$). Let
$V_{n}=\left\\{(x,y,z)\in\mathbb{F}_{p^{k_{n}}}^{3}:P(x)-P(y)=Q(z)\right\\}$
and $V=\prod_{n\to\mathcal{U}}{V_{n}}$. Finally, let $\mu_{V}$ be the Loeb
measure on $V$ obtained from the normalized counting measures on $V_{n}$.

Claim 1: $\mathbb{F}_{\infty}=\bigcup_{i=1}^{r}{C_{i}}$.

Let $x=(x_{n})_{n\in\mathbb{N}}\in\mathbb{F}_{\infty}$. For
$i\in\\{1,\dots,r\\}$, let $I_{i}=\left\\{n\in\mathbb{N}:x_{n}\in
C_{n,i}\right\\}$. Then $\mathbb{N}=\bigcup_{i=1}^{r}{I_{i}}$, so
$I_{i_{0}}\in\mathcal{U}$ for some $i_{0}\in\\{1,\dots,r\\}$, since
$\mathcal{U}$ is an ultrafilter. By the definition of the sets
$C_{1},\dots,C_{r}$, it follows that $x\in C_{i_{0}}$. This proves the claim.

Arguing as in the proof of Claim 1 above, one can check that $M$ is the set of
monochromatic solutions $(x,y,z)\in\mathbb{F}_{\infty}^{3}$ to the equation
$P(x)-P(y)=Q(z)$ with respect to the coloring
$\mathbb{F}_{\infty}=\bigcup_{i=1}^{r}{C_{i}}$.

Claim 2: $\mu_{V}(M)=0$.

We have constructed $M$ as an internal set, so by the definition of the Loeb
measure,

$\displaystyle\mu_{V}(M)=\lim_{n\to\mathcal{U}}{\frac{|M_{n}|}{|V_{n}|}}.$

Now, by Proposition 6.1,
$|V_{n}|=(p^{k_{n}})^{2}+O_{n\to\infty}\left((p^{k_{n}})^{2-\gamma}\right)$
for some $\gamma>0$. By assumption,
$|M_{n}|=o_{n\to\infty}\left((p^{k_{n}})^{2}\right)$. Hence,
$\frac{|M_{n}|}{|V_{n}|}=o_{n\to\infty}(1)$, so $\mu_{V}(M)=0$, since
$\mathcal{U}$ is non-principal.

Let $A_{i}=P(C_{i})=\prod_{n\to\mathcal{U}}{P(C_{n,i})}$. Note that

$\displaystyle\frac{1}{d}\mu(C_{i})\leq\mu(A_{i})\leq\mu(C_{i}),$

where $d=\deg{P}$. In particular, $\mu(A_{i})=0$ if and only if
$\mu(C_{i})=0$.

Without loss of generality, we may assume that $\mu(C_{i})>0$ for $1\leq i\leq
s$ and $\mu(C_{i})=0$ for $s+1\leq i\leq r$, for some $s\in\\{1,\dots,r\\}$.
Let $A=A_{1}\times\dots\times A_{s}\subseteq\mathbb{F}_{\infty}^{s}$. Note
that $\mu^{s}(A)=\prod_{i=1}^{s}{\mu(A_{i})}>0$. Let
$T_{z}:\mathbb{F}_{\infty}^{s}\to\mathbb{F}_{\infty}^{s}$ be the map
$T_{z}\mathbf{x}=(x_{1}+z,\dots,x_{s}+z)$ for $z\in\mathbb{F}_{\infty}$,
$\mathbf{x}=(x_{1},\dots,x_{s})\in\mathbb{F}_{\infty}^{s}$. For each
$n\in\mathbb{N}$ and $i\in\\{1,\dots,r\\}$, let $A_{n,i}=P(C_{n,i})$, and let
$A^{(n)}=A_{n,1}\times\dots\times A_{n,s}\in\mathbb{F}_{p^{k_{n}}}^{s}$. Also
let $T^{(n)}_{z}:\mathbb{F}_{p^{k_{n}}}^{s}\to\mathbb{F}_{p^{k_{n}}}^{s}$ be
the map $T^{(n)}_{z}\mathbf{x}=(x_{1}+z,\dots,x_{s}+z)$ for
$z\in\mathbb{F}_{p^{k_{n}}}$ and
$\mathbf{x}=(x_{1},\dots,x_{s})\in\mathbb{F}_{p^{k_{n}}}^{s}$. Now, since $Q$
is good for irrational equidistribution, we have

(6.1)
$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{\mathbf{x}\in\mathbb{F}_{p^{k_{n}}}^{s}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}_{p^{k_{n}}}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{Q(z)}\mathbf{x}\right)}}=\operatorname*{\text{\Large$\mathbb{E}$}}_{\mathbf{x}\in\mathbb{F}_{p^{k_{n}}}^{s}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}_{p^{k_{n}}}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{y}\mathbf{x}\right)}}+o_{n\to\infty}(1)$

by Proposition 6.3 and the Cauchy–Schwarz inequality. Hence,

$\displaystyle\int_{\mathbb{F}_{\infty}}{\mu^{s}\left(A\cap
T_{Q(z)}A\right)~{}d\mu(z)}$
$\displaystyle\stackrel{{\scriptstyle(1)}}{{=}}\int_{\mathbb{F}_{\infty}^{s+1}}{\mathbbm{1}_{A}(\mathbf{x})\mathbbm{1}_{A}(T_{Q(z)}\mathbf{x})~{}d\mu^{s+1}(\mathbf{x},z)}$
$\displaystyle\stackrel{{\scriptstyle(2)}}{{=}}\lim_{n\to\mathcal{U}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{(\mathbf{x},z)\in\mathbb{F}_{p^{k_{n}}}^{s+1}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{Q(z)}\mathbf{x}\right)}}$
$\displaystyle\stackrel{{\scriptstyle(3)}}{{=}}\lim_{n\to\mathcal{U}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{(\mathbf{x},y)\in\mathbb{F}_{p^{k_{n}}}^{s+1}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{y}\mathbf{x}\right)}}$
$\displaystyle\stackrel{{\scriptstyle(4)}}{{\geq}}\lim_{n\to\mathcal{U}}{\left(\frac{|A^{(n)}|}{p^{sk_{n}}}\right)^{2}}$
$\displaystyle\stackrel{{\scriptstyle(5)}}{{=}}\mu^{s}(A)^{2}>0.$

The steps are justified as follows. Step (1) is a direct application of
Proposition 6.5. The equality (2) comes from the definition of the Loeb
measure $\mu^{s+1}$. In step (3), we have taken the limit of both sides of
(6.1) along $\mathcal{U}$. The inequality (4) holds for each $n\in\mathbb{N}$
by the Cauchy–Schwarz inequality:

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{(\mathbf{x},y)\in\mathbb{F}_{p^{k_{n}}}^{s+1}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{y}\mathbf{x}\right)}$
$\displaystyle=\left\langle\mathbbm{1}_{A^{(n)}},\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}_{p^{k_{n}}}}{T^{(n)}_{y}\mathbbm{1}_{A^{(n)}}}\right\rangle$
$\displaystyle\geq\left\|\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in\mathbb{F}_{p^{k_{n}}}}{T^{(n)}_{y}\mathbbm{1}_{A^{(n)}}}\right\|_{L^{2}\left(\mathbb{F}_{p^{k_{n}}}^{s}\right)}^{2}$
$\displaystyle\geq\left(\frac{\left|A^{(n)}\right|}{p^{sk_{n}}}\right)^{2}.$

Finally, (5) follows from the definition of the Loeb measure $\mu^{s}$.

Thus,

$\displaystyle\mu\left(\left\\{z\in\mathbb{F}_{\infty}:\mu^{s}\left(A\cap
T_{Q(z)}A\right)>0\right\\}\right)>0.$

Let $G=\left\\{z\in\bigcup_{i=1}^{s}{C_{i}}:\mu^{s}\left(A\cap
T_{Q(z)}A\right)>0\right\\}$. Since the set $\bigcup_{i=s+1}^{r}{C_{i}}$ has
Loeb measure zero, $\mu(G)>0$.

For $z\in G$, let $i(z)\in\\{1,\dots,s\\}$ such that $z\in C_{i(z)}$. Noting
that

$\displaystyle\mu^{s}\left(A\cap
T_{Q(z)}A\right)=\prod_{i=1}^{s}{\mu\left(A_{i}\cap(A_{i}+Q(z))\right)},$

it follows that

$\displaystyle\mu\left(A_{i(z)}\cap\left(A_{i(z)}+Q(z)\right)\right)>0.$

The set $A_{i(z)}$ lies in the image of $P$ by definition, so taking the
inverse image under $P$,

$\displaystyle\mu\left(C_{i(z)}\cap
P^{-1}\left(A_{i(z)}+Q(z)\right)\right)>0.$

Therefore,

$\displaystyle\mu_{V}(M)$
$\displaystyle=\lim_{n\to\mathcal{U}}{\frac{|M_{n}|}{|V_{n}|}}$
$\displaystyle=\lim_{n\to\mathcal{U}}{\frac{|M_{n}|}{(p^{k_{n}})^{2}}}$
$\displaystyle=\lim_{n\to\mathcal{U}}{\frac{1}{(p^{k_{n}})^{2}}\sum_{i=1}^{r}{\sum_{z\in\mathbb{F}_{p^{k_{n}}}}{\mathbbm{1}_{C_{n,i}}(z)\left|\left\\{(x,y)\in
C_{n,i}^{2}:P(x)-P(y)=Q(z)\right\\}\right|}}}$
$\displaystyle\geq\lim_{n\to\mathcal{U}}{\sum_{i=1}^{r}{\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}_{p^{k_{n}}}}{\mathbbm{1}_{C_{n,i}}(z)\frac{\left|C_{n,i}\cap
P^{-1}\left(A_{n,i}+Q(z)\right)\right|}{p^{k_{n}}}}}}$
$\displaystyle=\sum_{i=1}^{r}{\int_{\mathbb{F}_{\infty}}{\mathbbm{1}_{C_{i}}(z)\mu\left(C_{i}\cap
P^{-1}\left(A_{i}+Q(z)\right)\right)}}$
$\displaystyle\geq\int_{G}{\mu\left(C_{i(z)}\cap
P^{-1}\left(A_{i(z)}+Q(z)\right)\right)~{}d\mu(z)}$ $\displaystyle>0.$

This final inequality contradicts Claim 2, so the theorem follows by _reductio
ad absurdum_. ∎

The proof of Theorem 1.19 is sufficiently flexible as to allow for many
variations. We note that in the proof of Theorem 1.19, we do not use the full
strength of the equidistribution assumption on $Q$. Carefully following each
step in the proof reveals the following characterization of polynomials
$Q(x)\in\mathbb{F}_{p}[x]$ for which the conclusion of Theorem 1.19 holds:

######  Corollary 6.7.

Let $Q(x)\in\mathbb{F}_{p}[x]$. Write
$Q(x)=Q(0)+\sum_{i=1}^{n}{\eta_{i}\left(x^{r_{i}}\right)}$ with
$r_{1},\dots,r_{n}$ distinct with $p\nmid r_{i}$ and
$\eta_{1},\dots,\eta_{n}\in\mathbb{F}_{p}[x]$ additive polynomials. For
$k\in\mathbb{N}$, let $H^{(k)}_{i}=\eta_{i}(\mathbb{F}_{p^{k}})$ and
$H^{(k)}=\sum_{i=1}^{n}{H^{(k)}_{i}}$. The following are equivalent:

  1. (i)

there exists $k_{0}\in\mathbb{N}$ such that $Q(0)\in H^{(k)}$ for all $k\geq
k_{0}$;

  2. (ii)

for any nonconstant polynomial $P(x)\in\mathbb{F}_{p}[x]$ and any
$r\in\mathbb{N}$, there exist $K=K(P,r)\in\mathbb{N}$ and $c(P,r)>0$ such that
for any $k\geq K$ and any $r$-coloring
$\mathbb{F}_{p^{k}}=\bigcup_{i=1}^{r}{C_{i}}$, there are at least
$c\left(p^{k}\right)^{2}$ monochromatic solutions to the equation
$P(x)-P(y)=Q(z)$.

######  Remark 6.8.

Condition (i) in Corollary 6.7 holds for intersective $Q$; see the first two
lines in the proof of Corollary 1.12.

###### Proof.

(i) $\implies$ (ii). The proof of Theorem 1.19 goes through with only minor
changes, which we will now describe. Equation (6.1) must be replaced with

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{\mathbf{x}\in\mathbb{F}_{p^{k_{n}}}^{s}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}_{p^{k_{n}}}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{Q(z)}\mathbf{x}\right)}}=\operatorname*{\text{\Large$\mathbb{E}$}}_{\mathbf{x}\in\mathbb{F}_{p^{k_{n}}}^{s}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in
H^{(k_{n})}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{y}\mathbf{x}\right)}}+o_{n\to\infty}(1).$

This follows from Proposition 6.3 together with the assumption (i) to
eliminate the presence of the constant term $Q(0)$. We then note that the
Cauchy–Schwarz inequality implies

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{\mathbf{x}\in\mathbb{F}_{p^{k_{n}}}^{s}}{\operatorname*{\text{\Large$\mathbb{E}$}}_{y\in
H^{(k_{n})}}{\mathbbm{1}_{A^{(n)}}(\mathbf{x})\mathbbm{1}_{A^{(n)}}\left(T^{(n)}_{y}\mathbf{x}\right)}}\geq\left(\frac{\left|A^{(n)}\right|}{p^{sk_{n}}}\right)^{2}$

for each $n\in\mathbb{N}$. The steps (3) and (4) in the proof of Theorem 1.19
may be replaced by the above considerations.

Finally, by Proposition 6.1, the set
$V_{n}=\left\\{(x,y,z)\in\mathbb{F}_{p^{k_{n}}}^{3}:P(x)-P(y)=Q(z)\right\\}$
has cardinality $|V_{n}|=\frac{(p^{k_{n}})^{3}}{|H_{n}|}$, where $H_{n}$ is
the subgroup generated by $H^{(k_{n})}$ and $\left\langle
P(x):x\in\mathbb{F}_{p^{k_{n}}}\right\rangle$. By assumption, $P$ is
nonconstant, so
$|H_{n}|\geq|P(\mathbb{F}_{p^{k_{n}}})|\geq\frac{p^{k_{n}}}{d}$, where
$d=\deg{P}$. Hence, $\left(p^{k_{n}}\right)^{2}\leq|V_{n}|\leq
d\left(p^{k_{n}}\right)^{2}$. This upper and lower bound on the cardinality of
$V_{n}$ allow for the remainder of the argument to be carried out without
difficulty.

(ii) $\implies$ (i). We prove the contrapositive. Suppose $Q(0)\notin
H^{(k)}$, and take $P=Q$. Then for any $x,y,z\in\mathbb{F}_{p^{k}}$, we have

$\displaystyle P(x)-P(y)-Q(z)$
$\displaystyle=\left(Q(0)+\sum_{i=1}^{n}{\eta_{i}\left(x^{r_{i}}\right)}\right)-\left(Q(0)+\sum_{i=1}^{n}{\eta_{i}\left(y^{r_{i}}\right)}\right)-\left(Q(0)+\sum_{i=1}^{n}{\eta_{i}\left(z^{r_{i}}\right)}\right)$
$\displaystyle\in-Q(0)+H^{(k)},$

so $P(x)-P(y)=Q(z)$ does not have any solutions over $\mathbb{F}_{p^{k}}$,
much less monochromatic solutions for an arbitrary coloring of
$\mathbb{F}_{p^{k}}$. ∎

Another feature of the proof of Theorem 1.19 is the following. Taking the
ultraproduct of a sequence of finite fields $\mathbb{F}_{q}$ with
characteristic growing to infinity, the same method shows that for any
nonconstant polynomials $P(x),Q(x)\in\mathbb{Z}[x]$, the equation
$P(x)-P(y)=Q(z)$ is partition regular over all fields of sufficiently high
characteristic. This follows by noting that $P$ and $Q$ will be nonconstant
and separable (hence good for irrational equidistribution; see Example 1.16(1)
above) once the characteristic exceeds the degrees of $P$ and $Q$ and the size
of some nonconstant coefficient.

In the special case $P=Q$, Theorem 1.19 can be seen as a polynomial version of
Schur’s theorem over finite fields. Indeed, the classical theorem of Schur
asserts that the equation $x+y=z$ is partition regular over $\mathbb{N}$. We
have just established partition regularity of the equation $P(x)+P(y)=P(z)$
over finite fields whenever $P$ satisfies the equidistribution assumption in
item (i) of Corollary 6.7. While the condition (i) in Corollary 6.7 depends on
the characteristic $p$, it is automatically satisfied for polynomials with
zero constant term. Hence, Corollary 1.20 holds.

The equation $P(x)-P(y)=Q(z)$ is often not partition regular (and may not even
be solvable) over $\mathbb{N}$. A key fact leveraged in the proof of Theorem
1.19 is that polynomials take on a positive proportion of values in finite
fields, something that is far from the case in $\mathbb{N}$. It remains an
interesting and difficult open problem, asked by Erdős and Graham in [EG80],
whether the Pythagorean equation $x^{2}+y^{2}=z^{2}$ is partition regular over
$\mathbb{N}$. (This was settled with a computer-assisted proof in the case of
2-colorings in [HKM16] but is wide open for 3 or more colors.)

Some comments are in order on the use of ultraproducts in the proofs of the
aforementioned partition regularity results. The basic strategy we have taken
is to discard those colors that have zero Loeb measure in the ultraproduct and
then to use recurrence along the polynomial $Q$ to find the desired points
$x,y,z$ with $P(x)-P(y)=Q(z)$. One may be tempted to carry out this strategy
in purely finitary terms, avoiding the use of ultraproducts and Loeb measure.
Unfortunately, this does not work (at least in its most straightforward
implementation). The following discussion illuminates the issues that arise.
Fix a polynomial $Q(x)\in\mathbb{F}_{p}[x]$ that is good for irrational
equidistribution. For simplicity, we will consider $P(x)=x$. Let
$r\in\mathbb{N}$. Suppose $k\in\mathbb{N}$ is large and an $r$-coloring
$\mathbb{F}_{p^{k}}=\bigcup_{i=1}^{r}{C_{i}}$ is given. We wish to use a
function $\Phi:\mathbb{N}\to\left[0,\frac{1}{r}\right]$ as a cutoff for
distinguishing “large” from “small” color classes. That is, we will consider a
color class $C_{i}$ large if $|C_{i}|\geq\Phi(k)p^{k}$ and small if
$|C_{i}|<\Phi(k)p^{k}$. Without loss of generality, we may assume
$C_{1},\dots,C_{s}$ are large and $C_{s+1},\dots,C_{r}$ are small for some
$s\in\\{1,\dots,r\\}$. (The requirement that $\Phi(k)\leq\frac{1}{r}$
guarantees that at least one color class is large.) We now proceed as in the
proof of Theorem 1.19, using “large” as a replacement for having positive Loeb
measure. Let $A=C_{1}\times\dots\times C_{s}$. Proposition 6.3 gives the bound

$\displaystyle\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}_{p^{k}}}{\frac{\left|A\cap\left(A+\left(Q(z),\dots,Q(z)\right)\right)\right|}{p^{sk}}}\geq\left(\frac{|A|}{p^{sk}}\right)^{2}+O_{k\to\infty}\left(p^{-\gamma
k}\right)$

where $\gamma=2^{-(\text{d-}\deg{Q-1)}}$. Since
$\left|A\cap\left(A+\left(Q(z),\dots,Q(z)\right)\right)\right|\leq|A|$ for
each $z\in\mathbb{F}_{p^{k}}$, we deduce that

$\displaystyle\left|\left\\{z\in\mathbb{F}_{p^{k}}:A\cap\left(A+\left(Q(z),\dots,Q(z)\right)\right)\neq\emptyset\right\\}\right|$
$\displaystyle\geq\frac{p^{(s+1)k}}{|A|}\operatorname*{\text{\Large$\mathbb{E}$}}_{z\in\mathbb{F}_{p^{k}}}{\frac{\left|A\cap\left(A+\left(Q(z),\dots,Q(z)\right)\right)\right|}{p^{sk}}}$
$\displaystyle\geq\frac{|A|}{p^{(s-1)k}}+O_{k\to\infty}\left(\frac{p^{(s+1)k}}{|A|}p^{-\gamma
k}\right)$
$\displaystyle\geq\Phi(k)^{s}p^{k}+O_{k\to\infty}\left(\Phi(k)^{-s}p^{(1-\gamma)k}\right),$

where in the last step we have used the bound $|A|\geq\Phi(k)^{s}p^{sk}$. In
order to complete the argument, we want to find $z\in\bigcup_{i=1}^{s}{C_{i}}$
satisfying $A\cap\left(A+\left(Q(z),\dots,Q(z)\right)\right)\neq\emptyset$. To
that end, one would like to show

$\displaystyle\left|\left\\{z\in\mathbb{F}_{p^{k}}:A\cap\left(A+\left(Q(z),\dots,Q(z)\right)\right)\neq\emptyset\right\\}\right|>\left|\bigcup_{i=s+1}^{r}{C_{i}}\right|.$

The total size of the small color classes is bounded by

$\displaystyle\left|\bigcup_{i=s+1}^{r}{C_{i}}\right|\leq(r-1)\Phi(k)p^{k}.$

The goal, then, is to choose the function
$\Phi:\mathbb{N}\to\left[0,\frac{1}{r}\right]$ so that

$\displaystyle\Phi(k)^{s}p^{k}>(r-1)\Phi(k)p^{k}+O_{k\to\infty}\left(\Phi(k)^{-s}p^{(1-\gamma)k}\right).$

Dividing by $p^{k}$, this reduces to the inequality

$\displaystyle\Phi(k)^{s}>(r-1)\Phi(k)+O_{k\to\infty}\left(\Phi(k)^{-s}p^{-\gamma
k}\right).$

But for $r\geq 2$, this requires $\Phi(k)>1$, which violates the condition
that $0\leq\Phi(k)\leq\frac{1}{r}$.

Working with the ultraproduct allows us to replace “small” with measure zero.
This is crucial, as we have just seen that “small” contributions in the
finitary setting may accumulate and overtake individual “large” terms. In
contrast, finite unions of measure zero sets remain of measure zero. However,
our infinitary methods come at a cost: we are unable to provide any
quantitative control on the values $K$ and $c$ appearing in the statement of
Theorem 1.19 and related corollaries. It is therefore an interesting problem
to obtain a purely finitary proof of Theorem 1.19 with effective bounds.

## Acknowledgements

The first author was supported by the National Science Foundation under Grant
No. DMS-1926686. We thank Peter Sarnak for pointing us to the work of Lang and
Weil [LW54] and for insightful discussions that helped shape Section 6.

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