${\textbf{v}}_{3}$ appearing as directions. However, this bound turns out to
be a key intermediate step for obtaining our claimed control of
$\Lambda(f_{0},f_{1},f_{2},f_{3})$ by a norm
$\left\|f_{3}\right\|_{{\textbf{v}}_{3}^{\times
s^{\prime}},{\mathbf{e}}_{3}^{\times s_{1}-1}}$ for some
$s^{\prime}\in{\mathbb{N}}$.

Step 2 (pong): Obtaining control by a norm of $f_{3}$.

In this step, our starting point is the newly obtained inequality (46).
Letting

$\displaystyle\tilde{f}_{2}=\operatorname*{\mathbb{E}}_{n}\overline{f_{0}}({\textbf{x}}-{\textbf{v}}_{2}n^{2})\overline{f_{1}}({\textbf{x}}+{\textbf{v}}_{1}n-{\textbf{v}}_{2}n^{2})\overline{f_{3}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)-{\textbf{v}}_{2}n^{2}),$

we employ Lemma A.1 to obtain

$\displaystyle|\Lambda(f_{0},f_{1},\tilde{f}_{2},f_{3})|\geq\delta^{2}.$

We then apply the inequality (46) with $\tilde{f}_{2}$ in place of $f_{2}$ to
conclude that

$\displaystyle\left\|\tilde{f}_{2}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1},{\textbf{v}}_{2}^{\times s_{2}}}\gg\delta^{O(1)},$

assuming like we have done so far that $\delta\gg p^{-c^{\prime}}$ for
$c^{\prime}>0$ sufficiently small. In the ping step, we applied Corollary A.3
in order to get rid just one vector ${\textbf{v}}_{3}-{\textbf{v}}_{2}$; this
was necessary for us to be able to apply the inverse theorem for degree 1
norm. This time, we aim at ridding ourselves of all the vectors
${\textbf{v}}_{2}^{\times s_{2}}$ at once since we do not want any such vector
to reappear in the norm of $f_{3}$ ultimately controlling our average.
Applying Corollary A.3(ii) this time, we get that

$\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})\cdot
f_{1,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{1}n)\cdot{\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})\cdot
f_{3,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))\mathclose{}\right|$

is at least $\Omega(\delta^{O(1)})$, where

$\displaystyle
f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=\Delta_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}})\quad\mathrm{for}\quad
j=0,1,3$

and ${\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}$ is a product of
$2^{s_{0}+s_{1}-1}$ elements of ${\mathfrak{D}}_{s_{2}}({\textbf{v}}_{2})$.
Let $B$ be the set of
$({\underline{h}},{\underline{h}}^{\prime})\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}$
for which

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})\cdot
f_{1,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{1}n)\cdot{\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})\cdot
f_{3,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))\mathclose{}\right|\gg\delta^{O(1)};$

we note that $|B|\gg\delta^{O(1)}p^{s_{0}+s_{1}-1}$ from the pigeonhole
principle. Proposition 4.5 tells us that in getting a box norm control of
counting operators twisted by dual functions, we can ignore the contribution
of dual functions. From Proposition 4.5 and the fact that the only quadratic
polynomial outside dual functions is in
$f_{3,{\underline{h}},{\underline{h}}^{\prime}}$ and lies along
${\textbf{v}}_{3}$, it follows that

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}1_{B}({\underline{h}},{\underline{h}}^{\prime})\cdot\left\|f_{3,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{U^{s_{3}}}\gg\delta^{O(1)}$

for some $s_{3}\in{\mathbb{N}}$. Extending by nonnegativity to all
$({\underline{h}},{\underline{h}}^{\prime})$, invoking the definition of
$f_{3,{\underline{h}},{\underline{h}}^{\prime}}$ and using the induction
formula for box norms, we deduce that

$\displaystyle\left\|f_{3}\right\|_{{\textbf{v}}_{3}^{\times
s^{\prime}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1}}\gg\delta^{O(1)}$

for $s^{\prime}=s_{0}+s_{3}$, which gives the claim. ∎

Proposition 5.1 can be used to control the other terms of the configuration by
Gowers norms as follows.

######  Proposition 5.3.

There exists $s\in{\mathbb{N}}$ and $c>0$ with the following property: for
every $D\in{\mathbb{N}}$ and 1-bounded functions
$f_{0},f_{1},f_{2},f_{3}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have the
bound

(47)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2},f_{3})|\ll\min_{j=1,2,3}\left\|f_{j}\right\|_{U^{s}({\textbf{v}}_{j})}^{c}+p^{-c}.$

###### Proof.

By Proposition 5.1, there exist $s_{1}\in{\mathbb{N}}$ and $c_{1}>0$ such that

(48)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2},f_{3})|\ll\left\|f_{3}\right\|_{U^{s_{1}}({\textbf{v}}_{3})}^{c_{1}}+p^{-c_{1}}$

for all 1-bounded functions
$f_{0},f_{1},f_{2},f_{3}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$. Suppose that
$\delta=|\Lambda(f_{0},f_{1},f_{2},f_{3})|$ for some $\delta\gg p^{-c}$ with
$0<c<c_{1}/2$. Letting

$\tilde{f}_{3}=\operatorname*{\mathbb{E}}_{n}\overline{f_{0}}({\textbf{x}}-{\textbf{v}}_{3}(n^{2}+n))\overline{f_{1}}({\textbf{x}}+{\textbf{v}}_{1}n-{\textbf{v}}_{3}(n^{2}+n))\overline{f_{2}}({\textbf{x}}+{\textbf{v}}_{2}n^{2}-{\textbf{v}}_{3}(n^{2}+n)),$

we obtain
$\left\|\tilde{f}_{3}\right\|_{U^{s_{1}}({\textbf{v}}_{3})}\gg\delta^{O(1)}$
from Lemma A.1. Another application of Lemma A.1, this time part (iii), gives

$\Lambda(f_{0},f_{1},f_{2},\overline{{\mathcal{D}}_{s_{1},{\textbf{v}}_{3}}\tilde{f}_{3}})\\\
=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2})\overline{{\mathcal{D}}_{s_{1},{\textbf{v}}_{3}}\tilde{f}_{3}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))\gg\delta^{O(1)}.$

We note that the expression above is the counting operator for the progression
${\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n,\;{\textbf{x}}+{\textbf{v}}_{2}n^{2}$
twisted by the term
$\overline{{\mathcal{D}}_{s_{1},{\textbf{v}}_{3}}\tilde{f}_{3}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))$
which can be removed using Proposition 4.5. It follows from this proposition
that there exists some $s_{2}\in{\mathbb{N}}$ such that - assuming $c>0$ is
small enough - we have
$\left\|f_{2}\right\|_{U^{s_{2}}({\textbf{v}}_{2})}\gg\delta^{O(1)}$.

We have thus shown that the counting operator is controlled by a Gowers norm
of $f_{2}$ and $f_{3}$, and it remains to prove a similar statement for
$f_{1}$. Letting

$\displaystyle\tilde{f}_{2}=\operatorname*{\mathbb{E}}_{n}\overline{f_{0}}({\textbf{x}}-{\textbf{v}}_{2}n^{2})\overline{f_{1}}({\textbf{x}}+{\textbf{v}}_{1}n-{\textbf{v}}_{2}n^{2}){\mathcal{D}}_{s_{1},{\textbf{v}}_{3}}\tilde{f}_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)-{\textbf{v}}_{2}n^{2})$

and applying Lemma A.1 as before, we deduce that

$\displaystyle\Lambda(f_{0},f_{1},\tilde{f}_{2},\overline{{\mathcal{D}}_{s_{1},{\textbf{v}}_{3}}\tilde{f}_{3}})\gg\delta^{O(1)}.$

Once again, Proposition 4.5 implies that
$\left\|\tilde{f}_{2}\right\|_{U^{s_{2}}({\textbf{v}}_{2})}\gg\delta^{O(1)}$.
From Lemma A.1(iii) we infer that

$\displaystyle\Lambda(f_{0},f_{1},\overline{{\mathcal{D}}_{s_{2},{\textbf{v}}_{2}}\tilde{f}_{2}},\overline{{\mathcal{D}}_{s_{1},{\textbf{v}}_{3}}\tilde{f}_{3}})\gg\delta^{O(1)}.$

A final application of Proposition 4.5 gives
$\left\|f_{1}\right\|_{U^{s_{3}}({\textbf{v}}_{1})}\gg\delta^{O(1)}$ for some
$s_{3}\in{\mathbb{N}}$, and the result follows by taking
$s=\max(s_{1},s_{2},s_{3})$. ∎

Before we move on to discuss the proof of Theorem 1.1 in the general case, we
describe certain reductions that happened in the proof of Proposition 5.2, and
which will shed light on how we prove Theorem 1.1 for general progressions. We
started with the task of controlling the counting operator

$\displaystyle\Lambda(f_{0},f_{1},f_{2},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))$

for the progression

${\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n,\;{\textbf{x}}+{\textbf{v}}_{2}n^{2},\;{\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n).$

In the ping step, we reduced this problem to one of controlling the counting
operator

$\displaystyle\Lambda^{\prime}(f_{0},f_{1},f_{2},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))$

for a progression

${\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n,\;{\textbf{x}}+{\textbf{v}}_{2}n^{2},\;{\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n);$

since both highest-degree polynomials lay along the same vector
${\textbf{v}}_{2}$, we could control the operator $\Lambda^{\prime}$ by
invoking Proposition 4.3. In the pong step, we similarly reduced to operators
of the form

$\displaystyle\Lambda(f_{0},f_{1},{\mathcal{D}},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n){\mathcal{D}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)),$

where ${\mathcal{D}}$ is a product of functions in
${\mathfrak{D}}({\textbf{v}}_{2})$; we can and will think of this as the
counting operator for the progression

$\displaystyle{\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n,\;{\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)$

twisted by the term ${\mathcal{D}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})$. This
time, the counting operator $\Lambda(f_{0},f_{1},{\mathcal{D}},f_{3})$ could
be controlled directly using Proposition 4.5. Lastly, in deriving Gowers norm
control on other terms of $\Lambda$ in Proposition 5.3, we reduced to the
operators

$\displaystyle\Lambda(f_{0},f_{1},f_{2},{\mathcal{D}})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2}){\mathcal{D}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)),$

and

$\displaystyle\Lambda(f_{0},f_{1},{\mathcal{D}}^{\prime},{\mathcal{D}})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n){\mathcal{D}}^{\prime}({\textbf{x}}+{\textbf{v}}_{2}n^{2}){\mathcal{D}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))$

respectively, where
${\mathcal{D}}\in{\mathfrak{D}}({\textbf{v}}_{3}),{\mathcal{D}}^{\prime}\in{\mathfrak{D}}({\textbf{v}}_{2})$,
both of which can be controlled directly using Proposition 4.5.

###  5.3. The formalism for longer progressions

To prove Theorem 1.2 in full generality, we need a robust formalism, imported
from our earlier ergodic work with Frantzikinakis [10]. We shall handle longer
families by reducing an arbitrary counting operator to a counting operator of
a smaller “type” much the same as we did in the proof of Proposition 5.2. In
what follows, we will be dealing with counting operators of the form

(49)
$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))$

for various choices of $\eta\in[\ell]^{\ell}$ and $\pi\in[L]^{L}$. We
encourage the reader to think of (49) as the counting operator for

$\displaystyle{\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{\eta_{1}}p_{1}(n),\;\ldots,\;{\textbf{x}}+{\textbf{v}}_{\eta_{\ell}}p_{\ell}(n)$

along 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, twisted by the
product
$\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))$,
where ${\mathcal{D}}_{j}\in{\mathfrak{D}}({\textbf{u}}_{\pi_{j}})$. In other
words, we do not think of
$\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))$
as an important part of the count (49), but rather as an annoying term that
can be removed using Proposition 4.4 and which therefore does not play a major
role in our analysis. We let

  * •

$\ell$ be the _length_ of (49) (noting that a counting operator of length
$\ell$ corresponds to a polynomial pattern of length $\ell+1$),

  * •

$d:=\max\limits_{j\in[\ell]}\deg p_{j}$ be its _degree_ ,

  * •

$\eta:=(\eta_{1},\ldots,\eta_{\ell})\in[\ell]^{\ell}$ be the _indexing tuple_
of (49).

Furthermore, we define

$\displaystyle{\mathfrak{L}}:=\\{j\in[\ell]\colon\ \deg p_{j}=d\\}$

to be the set of indices corresponding to polynomials $p_{j}$ of maximum
degree.

The relative complexity of various counting operators is measured by the
following notion of type. Letting $K=|{\mathfrak{L}}|$ be the number of
maximum degree polynomials $p_{1},\ldots,p_{\ell},$ we set the _type_ of (49)
to be the tuple $w:=(w_{1},\ldots,w_{\ell})$, where each entry $w_{t}$ is
defined by

$\displaystyle w_{t}:=|\\{j\in{\mathfrak{L}}\colon\
\eta_{j}=t\\}|=|\\{j\in[\ell]\colon\ \eta_{j}=t,\ \deg p_{j}=d\\}|;$

thus, it represents the number of polynomials $p_{j}$ of maximum degree
appearing along the vector ${\textbf{v}}_{t}$. We note that
$|w|:=w_{1}+\cdots+w_{\ell}=K$. For instance, the counting operator

$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}$ $\displaystyle
f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n^{2})f_{2}({\textbf{x}}+{\textbf{v}}_{2}n)f_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))$
$\displaystyle
f_{4}({\textbf{x}}+{\textbf{v}}_{1}(n^{2}+2n))f_{5}({\textbf{x}}+{\textbf{v}}_{3}(2n^{2}+n)){\mathcal{D}}({\textbf{x}}+{\textbf{u}}n^{3})$

for some ${\mathcal{D}}\in{\mathfrak{D}}({\textbf{u}})$ has length 5, degree
2, $K=4$ and indexing tuple $(1,2,3,1,3)$ \- we stress that we ignore the term
${\mathcal{D}}({\textbf{x}}+{\textbf{u}}n^{3})$ while discussing the
aforementioned parameters, hence the degree three polynomial $n^{3}$ and its
direction vector u is ignored. This counting operator has type $(2,0,2,0,0)$
since two of the quadratic polynomials lie along ${\textbf{v}}_{1}$ and two
others lie along ${\textbf{v}}_{3}$.

To organise the induction scheme, we need to define an ordering on types
$w\in[0,K]^{\ell}$. Let $\textrm{supp}(w)=\\{t\in[0,K]:\ w_{t}>0\\}$. For
distinct integers $m,i\in[\ell]$ with $m\in\textrm{supp}(w)$, we define the
type operation

$\displaystyle(\sigma_{mi}w)_{t}:=\begin{cases}w_{t},\;&t\neq m,i\\\
w_{m}-1,\;&t=m\\\ w_{i}+1,\;&t=i.\end{cases}.$

For instance, $\sigma_{12}(2,3,7)=(1,4,7)$. Letting
$w^{\prime}:=\sigma_{mi}w$, we set $w^{\prime}<w$ if $w_{m}\leq w_{i}$. In
particular, $(1,4,7)<(2,3,7)$ in the example above. We note that the tuple
$(2,3,7)$ of higher type has smaller variance than the tuple $(1,4,7)$ of
smaller type; this is a consequence of the fact that while passing from
$(2,3,7)$ to $(1,4,7)$, we decrease by 1 the smallest nonzero value 2. This
observation carries forward more generally: if the condition $w_{m}\leq w_{i}$
is satisfied, then an easy computation shows that $w^{\prime}$ has strictly
higher variance than $w$, or equivalently
${w^{\prime}_{1}}^{2}+\cdots+{w^{\prime}_{K}}^{2}>w_{1}^{2}+\cdots+w_{\ell}^{2}$.
Thanks to this fact, we can extend the ordering $<$ to all tuples
$[0,K]^{\ell}$ of length $K$ by transitivity, and so for two type tuples
$w,w^{\prime}\in[0,K]^{\ell}$, we let $w^{\prime}<w$ if there exist types
$w_{0},\ldots,w_{r}$ with $w_{0}=w$, $w_{r}=w^{\prime}$, such that for every
$l=0,\ldots,r-1$, we have $w_{l+1}=\sigma_{mi}w_{l}$ for distinct
$m,i\in{\mathfrak{L}}$ with $w_{lm}\leq w_{li}$. For instance, this ordering
induces the following chains of types:

$\displaystyle(4,0,0)<(3,1,0)<(2,2,0)<(2,1,1)\quad\textrm{and}\quad(0,4,0)<(1,3,0)<(2,2,0)<(2,1,1).$

We note here that the highest order type $w\in[0,K]^{\ell}$ of length $K$ is
one whose entries only take values 0 and 1, corresponding to the operators
(49) in which each highest degree polynomial $p_{j}$ lies along a different
vector. By contrast, the lowest order type is one in which one entry is $K$
while the other ones are all 0; this corresponds to the counting operator in
which all highest degree polynomials $p_{j}$ lie along the same direction.
Theorem 1.3 for such operators is a direct consequence of Proposition 4.5. We
call the types of the latter form basic. Thus, the ordering on types expresses
the intuition that counting operators with plenty of different direction
vectors are more complex than those with few.

Our induction scheme works as follows. We want to obtain Gowers norm control
on general counting operators of length $\ell$ and type $w$ of the form (49)
in which the polynomials $p_{1},\ldots,p_{\ell}$ are pairwise independent. For
instance, the counting operator

$\displaystyle\Lambda(f_{0},f_{1},f_{2},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))$

that we examined in Proposition 5.2 has length $\ell=3$ and type $w=(0,1,1)$.
In the ping step, we will reduce the counting operator to one of the same
length $\ell$ but lower type $w^{\prime}=\sigma_{mi}w$ for some $m,i$ with
$w_{m}<w_{i}$; this will correspond to replacing one instance of the vector
${\textbf{v}}_{m}$ by the vector ${\textbf{v}}_{i}$. Indeed, in the ping step
of Proposition 5.2, we reduced to the counting operator

$\displaystyle\Lambda^{\prime}(f_{0},f_{1},f_{2},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))$

of the same length $\ell=3$ but lower type
$w^{\prime}=\sigma_{32}w=(0,2,0)<w$. The type is basic, and we indeed obtained
the control over $\Lambda^{\prime}$ by directly invoking Proposition 4.5 (in
that case, it sufficed to invoke Proposition 4.3). In the pong step, we will
reduce the counting operator to one of smaller length $\ell-1$; this will
correspond to replacing one of the terms by a product of dual functions. In
Proposition 5.2, the relevant operator of shorter length 2 took the form

$\displaystyle\Lambda(f_{0},f_{1},{\mathcal{D}},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n){\mathcal{D}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)).$

Lastly, once we obtain Gowers norm control on one of the terms with highest-
degree polynomials, we will use the weak inverse theorem for Gowers norms to
replace this term in the counting operator by a dual function, once again
reducing the counting operator to one of length $\ell-1$, to which we will
apply the induction hypothesis in order to get Gowers norm control for other
terms. In Proposition 5.3, we did this by passing to the counting operator

$\displaystyle\Lambda(f_{0},f_{1},f_{2},{\mathcal{D}})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2}){\mathcal{D}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n)).$

of length $2$.

Thus, in the ping step, we induct on $<$ for counting operators of the same
length $\ell$; whereas in the pong step and while extending Gowers norm
control from one term to all other terms, we induct on the length $\ell$.
Reducing the counting operators to those of lower length, we will arrive after
finitely many steps at counting operators of length $1$, for which we have
Gowers norm control by Proposition 4.5. Similarly, reducing the counting
operators in the ping step to those of the same length and lower type, we will
arrive after finitely many steps at averages of length $\ell$ and basic type,
for which Gowers norm control also follows from Proposition 4.5.

###  5.4. The induction scheme

We now present three propositions whose interplay will give Theorem 1.3. The
first of them gives Gowers norm control for a general class of twisted
counting operators. Theorem 1.3 follows from the result below upon setting
$L:=0$ and letting $\eta$ be the identity indexing tuple.

######  Proposition 5.4.

Let $d,D,\ell\in{\mathbb{N}}$, $L\in{\mathbb{N}}_{0}$,
$\eta\in[\ell]^{\ell},\pi\in[L]^{L}$ be indexing tuples,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell},{\textbf{u}}_{1},\ldots,{\textbf{u}}_{L}\in{\mathbb{Z}}^{D}$
be nonzero vectors and
$p_{1},\ldots,p_{\ell},q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials of
degree at most $d$ with zero constant terms such that $p_{1},\ldots,p_{\ell}$
are pairwise independent. Then there exists $s^{\prime}=O_{d,\ell,L}(1)$ such
that for all 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ and
${\mathcal{D}}_{j}\in{\mathfrak{D}}_{d}({\textbf{u}}_{\pi_{j}})$, we have

(50)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|^{O_{d,\ell,L}(1)}\\\
\ll_{d,\ell,L}\min_{j\in[\ell]}\left\|f_{j}\right\|_{U^{s}({\textbf{v}}_{\eta_{j}})}+p^{-1}.$

Proposition 5.4 will be deduced from the following result.

######  Proposition 5.5.

Let $d,D,\ell\in{\mathbb{N}}$, $L\in{\mathbb{N}}_{0}$,
$\eta\in[\ell]^{\ell},\pi\in[L]^{L}$ be indexing tuples,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell},{\textbf{u}}_{1},\ldots,{\textbf{u}}_{L}\in{\mathbb{Z}}^{D}$
be nonzero vectors and
$p_{1},\ldots,p_{\ell},q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials of
degree at most $d$ with zero constant terms such that $p_{1},\ldots,p_{\ell}$
are pairwise independent. Suppose that the type $w$ of the counting operator
(49) is not basic, and let $m\in[\ell]$ be such that
$w_{\eta_{m}}=\min\limits_{t\in\textrm{supp}(w)}w_{t}$. Then there exist
$s=O_{d,\ell,L}(1)$ such that for all 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ and
${\mathcal{D}}_{j}\in{\mathfrak{D}}_{d}({\textbf{u}}_{\pi_{j}})$, we have

(51)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|^{O_{d,\ell,L}(1)}\\\
\ll_{d,\ell,L}\left\|f_{m}\right\|_{U^{s}({\textbf{v}}_{\eta_{m}})}+p^{-1}.$

For instance, if $w:=(1,2,1,0,0,3,0)$, then (51) holds for $m\in[7]$
satisfying $\eta_{m}=1$ or $\eta_{m}=3$.

Proposition 5.5 is a straightforward consequence of Proposition 4.5, followed
by an iterated application of the smoothing result given below.

######  Proposition 5.6.

Let $d,D,\ell,s_{0},s_{1}\in{\mathbb{N}}$, $L\in{\mathbb{N}}_{0}$,
$\eta\in[\ell]^{\ell},\pi\in[L]^{L}$ be indexing tuples,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell},{\textbf{u}}_{1},\ldots,{\textbf{u}}_{L}\in{\mathbb{Z}}^{D}$
be nonzero vectors and
$p_{1},\ldots,p_{\ell},q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials of
degree at most $d$ with zero constant terms such that $p_{1},\ldots,p_{\ell}$
are pairwise independent. Suppose that the type $w$ of the counting operator
(49) is not basic, and let $m\in[\ell]$ be such that
$w_{\eta_{m}}=\min\limits_{t\in\textrm{supp}(w)}w_{t}$. Then for each vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}}$ satisfying (30) which are not
scalar multiples of the vector ${\textbf{v}}_{\eta_{m}}$, there exists
$s=O_{d,\ell,L,s_{0},s_{1}}(1)$ with the following property: for all 1-bounded
functions $f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}$ and
${\mathcal{D}}_{j}\in{\mathfrak{D}}_{d}({\textbf{u}}_{\pi_{j}})$, the bound

(52)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|^{O_{d,\ell,L}(1)}\\\
\ll_{d,\ell,L}\left\|f_{m}\right\|_{{\textbf{v}}_{\eta_{m}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}}}+p^{-1}.$

implies

(53)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|^{O_{d,\ell,L}(1)}\\\
\ll_{d,\ell,L}\left\|f_{m}\right\|_{{\textbf{v}}_{\eta_{m}}^{\times
s^{\prime}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1}}+p^{-1}.$

In other words, Proposition 5.6 tells that one by one, we can replace all the
directions different from ${\textbf{v}}_{\eta_{m}}$ in the box norm of $f_{m}$
controlling the counting operator by many copies of ${\textbf{v}}_{\eta_{m}}$
until we obtain control by a large degree Gowers norm along
${\textbf{v}}_{\eta_{m}}$.

The relationship between Propositions 5.4 \- 5.6 is as follows:

  * •

for counting operators of length $\ell$ and basic type $w$, Proposition 5.4
follows directly from Proposition 4.5 (in particular, this includes the case
when the counting operator has length $\ell=1$);

  * •

for counting operators of length $\ell$ and non-basic type $w$, Proposition
5.4 follows from Proposition 5.5 for $\ell$ and $w$ (which in turn follows
from an iterated application of Proposition 5.6 for this data), as well as
Proposition 5.4 for operators of length $\ell-1$;

  * •

for counting operators of length $\ell$ and non-basic type $w$, Proposition
5.6 follows from Proposition 5.4 for length $\ell$ and types $w^{\prime}<w$ as
well as length $\ell-1$.

###### Proof of Proposition 5.6.

We prove Proposition 5.6 for a counting operator (49) of length $\ell$ and
type $w$ by assuming that Proposition 5.4 holds for operators of length
$\ell-1$ as well as those of length $\ell$ and type $w^{\prime}<w$. For
simplicity of notation, we assume $m=\ell$.

By Proposition 4.5 and the assumption that the polynomials
$p_{1},\ldots,p_{\ell}$ are distinct and have zero constant terms, the vector
${\textbf{b}}_{s_{1}}$ is nonzero and equals
${\textbf{b}}_{s_{1}}=b_{\ell}{\textbf{v}}_{\eta_{\ell}}-b_{i}{\textbf{v}}_{\eta_{i}}$
for some $b_{\ell},b_{i}\in{\mathbb{Z}}$ with $b_{\ell}\neq 0$ and
$i\in\\{0,\ldots,\ell-1\\}$. By a further assumption, the vector
${\textbf{b}}_{s_{1}}$ is not a scalar multiple of
${\textbf{v}}_{\eta_{\ell}}$ (in particular, $i\neq 0$), and so
$\eta_{\ell}\neq\eta_{i}$ and $b_{i}\neq 0$. The proof of Proposition 5.6
follows the same two-step strategy as the proof of Proposition 5.2. We first
obtain the control of (49) by
$\left\|f_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s_{1}-1}},{\textbf{v}}_{\eta_{i}}^{\times
s_{2}}}$ for some $s_{2}=O_{d,\ell,L,s_{0},s_{1}}(1)$. This is accomplished by
using the control by
$\left\|f_{\ell}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}}}$, given by assumption,
for an appropriately defined function $\tilde{f}_{\ell}$ in place of
$f_{\ell}$. Subsequently, we repeat the procedure by applying the newly
established control by
$\left\|f_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s_{1}-1}},{\textbf{v}}_{\eta_{i}}^{\times
s_{2}}}$ for a function $\tilde{f}_{i}$ in place of $f_{i}$. This gives us the
claimed result.

Step 1 (ping): Obtaining control by a norm of $f_{i}$.

Fix $f_{0},\ldots,f_{\ell}$, ${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}$ and
let

$\displaystyle\delta=\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|.$

Assume without the loss of generality that $\delta\gg p^{c}$ for a
sufficiently small $c>0$. We let $c$ and all the other constants in this proof
depend on $d,\ell,L,s_{0},s_{1}$.

We set $p_{0}=0$, $\eta_{0}=0$ and ${\textbf{v}}_{0}=\mathbf{0}$. Defining

$\displaystyle\tilde{f}_{\ell}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\prod_{j=0}^{\ell-1}\overline{f_{j}}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n)-{\textbf{v}}_{\eta_{\ell}}p_{\ell}(n))\cdot\prod_{j\in[L]}\overline{{\mathcal{D}}_{j}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n)-{\textbf{v}}_{\eta_{\ell}}p_{\ell}(n)),$

we deduce from Lemma A.1 that

$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{j=0}^{\ell-1}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\tilde{f}_{\ell}({\textbf{x}}+{\textbf{v}}_{\eta_{\ell}}p_{\ell}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\geq\delta^{2}.$

Then our assumption gives

$\left\|\tilde{f}_{\ell}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}}}\gg\delta^{O(1)}.$

By Corollary A.3, the expression

$\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{j=0}^{\ell-1}f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta_{\ell}}p_{\ell}(n))\cdot\prod_{j=1}^{L}{\mathcal{D}}^{\prime}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|$

has size $\Omega(\delta^{O(1)})$, where

$\displaystyle f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$
$\displaystyle=\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}})\quad\textrm{for}\quad
j\in\\{0,\ldots,\ell-1\\}$
$\displaystyle{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$
$\displaystyle=\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}{\mathcal{D}}_{j}({\textbf{x}})\quad\textrm{for}\quad
j\in[L].$

In particular, each
${\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}$ is a product of
$2^{s_{0}+s_{1}-1}$ elements of ${\mathfrak{D}}_{d}({\textbf{u}}_{\pi_{j}})$.
The functions $u_{{\underline{h}},{\underline{h}}^{\prime}}$ appearing in the
expression above are 1-bounded and invariant under
${\textbf{b}}_{s_{1}}=b_{\ell}{\mathbf{e}}_{\eta_{\ell}}-b_{i}{\mathbf{e}}_{\eta_{i}}$
for some $i\in{\mathfrak{L}}$ such that $\eta_{\ell}\neq\eta_{i}$ and
$b_{\ell},b_{i}\neq 0$. This invariance property implies that

(54)
$u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+b_{\ell}{\textbf{v}}_{\eta_{\ell}}n)=u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+b_{i}{\textbf{v}}_{\eta_{i}}n)$

for every $n\in{\mathbb{F}}_{p}$. Setting

$\displaystyle p^{\prime}_{j}(n)$
$\displaystyle:=\begin{cases}p_{j}(n),\;&j\in\\{0,\ldots,\ell-1\\},\\\
\frac{b_{i}}{b_{\ell}}p_{\ell}(n),&j=\ell,\end{cases}$
$\displaystyle\eta^{\prime}_{j}$
$\displaystyle:=\begin{cases}\eta_{j},\;&j\in\\{0,\ldots,\ell-1\\},\\\
\eta_{i},\;&j=\ell,\end{cases}$

we deduce from the invariance property that

$\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{j=0}^{\ell-1}f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta^{\prime}_{j}}p^{\prime}_{j}(n))\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta^{\prime}_{\ell}}p^{\prime}_{\ell}(n))\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|$

has size $\Omega(\delta^{O(1)})$. We subsequently employ the pigeonhole
principle to find a set $B\subset{\mathbb{F}}_{p}^{2(s_{0}+s_{1}-1)}$ of
cardinality $|B|\gg\delta^{O(1)}p^{2(s_{0}+s_{1}-1)}$ such that for each
$({\underline{h}},{\underline{h}}^{\prime})\in B$, we have

(55)
$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{j=0}^{\ell-1}f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta^{\prime}_{j}}p^{\prime}_{j}(n))\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta^{\prime}_{\ell}}p^{\prime}_{\ell}(n))\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|\gg\delta^{O(1)}.$

Since the polynomials $p_{1},\ldots,p_{\ell}$ are pairwise independent, so are
$p^{\prime}_{1},\ldots,p^{\prime}_{\ell}$. While pairwise independence is
preserved under scaling, distinctness is not, and so in order to carry out the
argument inductively, we need the polynomials to be pairwise independent
rather than distinct.

We recall the assumptions that $w_{\eta_{\ell}}$ minimises
$(w_{t})_{t\in\textrm{supp}(w)}$ and $\eta_{\ell}\neq\eta_{i}$. The structure
of the set (30) implies that $w_{\eta_{i}}>0$, and so $w_{\eta_{i}}\geq
w_{\eta_{\ell}}$. Therefore, the type
$w^{\prime}=\sigma_{\eta_{\ell}\eta_{i}}w$ of the counting operator (55) is
strictly smaller than $w$. We inductively apply Proposition 5.4 to find
$s_{2}\in{\mathbb{N}}$ such that for every
$({\underline{h}},{\underline{h}}^{\prime})\in B$, we have

$\left\|f_{i,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{U^{s_{2}}({\textbf{v}}_{\eta_{i}})}=\left\|\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{i}\right\|_{U^{s_{2}}({\textbf{v}}_{\eta_{i}})}\gg\delta^{O(1)}.$

The nonnegativity of the box norms and the bound on the size of $B$ yield the
lower bound

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\left\|\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{i}\right\|_{U^{s_{2}}({\textbf{v}}_{\eta_{i}})}\gg\delta^{O(1)}.$

Replacing the difference ${\underline{h}}-{\underline{h}}^{\prime}$ by
${\underline{h}}$ and using the inductive formula for box norms together with
Hölder’s inequality, we conclude that

$\left\|f_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1},{\textbf{v}}_{\eta_{i}}^{\times
s_{2}}}\gg\delta^{O(1)},$

and so the norm $\left\|f_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1},{\textbf{v}}_{\eta_{i}}^{\times
s_{2}}}$ controls the counting operator (49) in the sense that

(56)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{j=0}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|^{O(1)}\\\
\ll\left\|f_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1},{\textbf{v}}_{\eta_{i}}^{\times
s_{2}}}+p^{-1}.$

Step 2 (pong): Obtaining control by a norm of $f_{\ell}$.

To get the claim that
$\left\|f_{\ell}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s^{\prime}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1}}$ controls the
average for some $s^{\prime}\in{\mathbb{N}}$, we repeat the procedure once
more with $f_{i}$ in place of $f_{\ell}$. Letting

$\displaystyle\tilde{f}_{i}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
i\end{subarray}}\overline{f_{j}}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n)-{\textbf{v}}_{\eta_{i}}p_{i}(n))$
$\displaystyle\prod_{j\in[L]}\overline{{\mathcal{D}}_{j}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n)-{\textbf{v}}_{\eta_{i}}p_{i}(n)),$

we deduce from (56) and Lemma A.1 that

$\displaystyle\left\|\tilde{f}_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1},{\textbf{v}}_{\eta_{i}}^{\times
s_{2}}}\gg\delta^{O(1)}.$

We want to get rid of all the $s_{2}$ vectors ${\textbf{v}}_{\eta_{i}}$ from
the norm. Corollary A.2 implies that

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
i\end{subarray}}f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot{\mathcal{D}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta_{i}}p_{i}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|$

is at least $\Omega(\delta^{O(1)})$, where

$\displaystyle f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$
$\displaystyle=\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}})\quad\textrm{for}\quad
j\in[0,\ell]\setminus\\{i\\},$
$\displaystyle{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$
$\displaystyle=\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}{\mathcal{D}}_{j}({\textbf{x}})\quad\textrm{for}\quad
j\in[L].$

Thus, each ${\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}$ is a
product of $2^{s_{0}+s_{1}-1}$ elements of
${\mathfrak{D}}_{d}({\textbf{u}}_{\pi_{j}})$. The functions
${\mathcal{D}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}$ appearing
in the expression above are products of $2^{s_{0}+s_{1}-1}$ elements of
${\mathfrak{D}}_{s_{2}}({\textbf{v}}_{\eta_{i}})$.

By the pigeonhole principle, there exists a set
$B^{\prime}\subset{\mathbb{F}}_{p}^{2(s_{0}+s_{1}-1)}$ with
$|B^{\prime}|\gg\delta^{O(1)}p^{2(s_{0}+s_{1}-1)}$ such that for every
$({\underline{h}},{\underline{h}}^{\prime})\in B^{\prime}$, the expression

(57)
$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
i\end{subarray}}f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot{\mathcal{D}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{\eta_{i}}p_{i}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|$

is at least $\Omega(\delta^{O(1)})$. Importantly, each of the averages in (57)
has length $\ell-1$ since the term $f_{i}$ has been replaced by a product of
dual functions. We therefore apply Proposition 5.4 inductively to find
$s_{3}\in{\mathbb{N}}$ such that

$\displaystyle\left\|f_{\ell,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{U^{s_{3}}({\textbf{v}}_{\eta_{\ell}})}=\left\|\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{\ell}\right\|_{U^{s_{3}}({\textbf{v}}_{\eta_{\ell}})}\gg\delta^{O(1)}$

for every $({\underline{h}},{\underline{h}}^{\prime})\in B^{\prime}$.

The nonnegativity of the box norms and the bound on the size of $B^{\prime}$
yield the lower bound

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\left\|\Delta_{{\textbf{v}}_{\eta_{\ell}}^{\times
s_{0}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{\ell}\right\|_{U^{s_{3}}({\textbf{v}}_{\eta_{\ell}})}\gg\delta^{O(1)}.$

Replacing ${\underline{h}}-{\underline{h}}^{\prime}$ by ${\underline{h}}$ and
applying the inductive formula for box norms alongside the Hölder inequality,
we conclude that

$\left\|f_{i}\right\|_{{\textbf{v}}_{\eta_{\ell}}^{\times
s^{\prime}},{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s_{1}-1}}\gg\delta^{O(1)}$

for $s^{\prime}=s_{0}+s_{3}$. ∎

Finally, we prove Proposition 5.4 for averages of length $\ell$ and type $w$.

###### Proof of Proposition 5.4.

When the type of the counting operator is basic, Proposition 5.4 follows
directly from Proposition 4.5. In particular, this includes the case $\ell=1$.
We assume therefore that $\ell>1$ and the type $w$ is not basic. It follows
from Proposition 5.5 that there exist $m\in[\ell]$ and $s_{1}=O_{d,\ell,L}(1)$
such that (51) holds with $s_{1}$ in place of $s$. Fix
$f_{0},\ldots,f_{\ell},$ ${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}$ and
suppose that

$\displaystyle\delta=\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\cdot\prod_{j=0}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|$

satisfies $\delta\gg p^{c}$ for an appropriately small $c>0$. Letting

$\displaystyle\tilde{f}_{m}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
m\end{subarray}}\overline{f_{j}}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n)-{\textbf{v}}_{\eta_{m}}p_{m}(n))\prod_{j\in[L]}\overline{{\mathcal{D}}_{j}}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n)-{\textbf{v}}_{\eta_{m}}p_{m}(n)),$

we deduce from Lemma A.1 that

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{\begin{subarray}{c}j\in[0,\ell]\setminus\\{m\\}\end{subarray}}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot\tilde{f}_{m}({\textbf{x}}+{\textbf{v}}_{\eta_{m}}p_{m}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|\geq\delta^{2}.$

Proposition 5.5 then gives
$\left\|\tilde{f}_{m}\right\|_{U^{s}({\textbf{v}}_{\eta_{m}})}\gg\delta^{O(1)}$,
and Lemma A.1 further implies that

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{\begin{subarray}{c}j\in[0,\ell]\setminus\\{m\\}\end{subarray}}f_{j}({\textbf{x}}+{\textbf{v}}_{\eta_{j}}p_{j}(n))\cdot{\mathcal{D}}_{s,{\textbf{v}}_{\eta_{m}}}\tilde{f}_{m}({\textbf{x}}+{\textbf{v}}_{\eta_{m}}p_{m}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{\pi_{j}}q_{j}(n))\mathclose{}\right|$

is at least $\Omega(\delta^{O(1)})$. This average has length $\ell-1$, and so
we apply Proposition 5.4 to find $s_{2}=O_{d,\ell,L}(1)$ such that
$\left\|f_{j}\right\|_{U^{s_{2}}({\textbf{v}}_{\eta_{j}})}\gg\delta^{O(1)}$
for all $j\in[\ell]\setminus\\{m\\}$. The result follows by setting
$s=\max(s_{1},s_{2})$. ∎

##  6\. Degree lowering

Last but not least, we derive Theorem 1.2 from Theorem 1.3. As stated in the
introduction, Theorem 1.2 can be derived from Theorem 1.3 with the exact same
degree lowering argument that was used to derive [19, Theorem 1.2], the
special case of Theorem 1.2 for distinct degree polynomials, from [19,
Proposition 4.1]. Here, we give an alternative version of the degree lowering
argument, which we believe gives a better conceptual understanding of what
really is going on.

A key concept that we will need is that of an eigenfunction. For
${\textbf{v}}\in{\mathbb{Z}}^{d}$, we say that
$\chi:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ is an eigenfunction of v with
eigenvalue $\phi:{\mathbb{F}}_{p}^{D}\to{\mathbb{F}}_{p}$ if:

  1. (i)

(Modulus) $|\chi({\textbf{x}})|\in\\{0,1\\}$ for every
${\textbf{x}}\in{\mathbb{F}}_{p}^{D}$;

  2. (ii)

(Eigenfunction property)
$\chi({\textbf{x}}+{\textbf{v}})=e_{p}(\phi({\textbf{x}}))\chi({\textbf{x}})$;

  3. (iii)

(Invariance) $\phi$ is v-invariant.

More explicitly, if $H=\langle{\textbf{v}}\rangle^{\perp}$ and
${\textbf{x}}={\textbf{x}}^{\prime}+{\textbf{v}}n$ is the unique decomposition
of x into two parts ${\textbf{x}}^{\prime}\in H$ and
${\textbf{v}}n\in\langle{\textbf{v}}\rangle$, then
$\chi({\textbf{x}})=\chi({\textbf{x}}^{\prime}+{\textbf{v}}n)=1_{E}({\textbf{x}}^{\prime})\lambda
e_{p}(\phi({\textbf{x}}^{\prime})n+\psi({\textbf{x}}^{\prime}))$ for some
$\phi,\psi:H\to{\mathbb{F}}_{p}$, $|\lambda|=1$ and $E\subset H$.

We let ${\mathcal{E}}({\textbf{v}})$ denote the group of eigenfunctions of v;
we note that it is closed under complex conjugation. These eigenfunctions are
a relatively simple example of the notion of nonergodic eigenfunctions
originally defined in ergodic setting by Frantzikinakis and Host [9]. The name
comes from the fact that on each coset of $\langle{\textbf{v}}\rangle$, our
eigenfunctions become “classical eigenfunctions” with respect to the
transformation $T{\textbf{x}}={\textbf{x}}+{\textbf{v}}$.

For instance, if $D=2$ and ${\textbf{v}}=(1,0)$, each eigenfunction
$\chi\in{\mathcal{E}}({\textbf{v}})$ takes the form

$\displaystyle\chi(x_{1},x_{2})=\lambda\cdot 1_{E}(x_{2})\cdot
e_{p}(\phi(x_{2})x_{1}+\psi(x_{2}))\quad\textrm{for}\quad|\lambda|=1\quad\textrm{and}\quad
E\subset{\mathbb{F}}_{p}.$

If $\chi\in{\mathcal{E}}({\textbf{v}})$ is an eigenfunction with eigenvalue
$\phi$, then

(58)
$\displaystyle\operatorname*{\mathbb{E}}(\chi|{\textbf{v}})({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\chi({\textbf{x}}+{\textbf{v}}n)=\chi({\textbf{x}})\operatorname*{\mathbb{E}}_{n}e_{p}(\phi({\textbf{x}})n)=\chi({\textbf{x}})1_{\phi({\textbf{x}})=0}.$

We will need this fact later.

With these definitions, the strong inverse theorem for $U^{2}({\textbf{v}})$
takes the following form.

######  Lemma 6.1 (Strong inverse theorem for $U^{2}({\textbf{v}})$).

Let $D\in{\mathbb{N}}$ and suppose that
$f:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ is 1-bounded. Then

$\displaystyle\left\|f\right\|_{U^{2}({\textbf{v}})}^{4}\leq\operatorname*{\mathbb{E}}_{{\textbf{x}}}f({\textbf{x}})\chi({\textbf{x}})$

for some $\chi\in{\mathcal{E}}({\textbf{v}})$ which additionally satisfies
$\operatorname*{\mathbb{E}}(f\cdot\chi|{\textbf{v}})\geq 0$ and $|\chi|=1$.

###### Proof.

Let $H=\langle{\textbf{v}}\rangle^{\perp}$, so that each
${\textbf{x}}\in{\mathbb{F}}_{p}^{D}$ can be uniquely decomposed as
${\textbf{x}}={\textbf{x}}^{\prime}+{\textbf{v}}n$ for
${\textbf{x}}^{\prime}\in H$ and $n\in{\mathbb{F}}_{p}$. For each
${\textbf{x}}^{\prime}\in H$ and $n\in{\mathbb{F}}_{p}$, we then define
$f_{{\textbf{x}}^{\prime}}(n)=f({\textbf{x}}^{\prime}+{\textbf{v}}n)$. Then
the usual $U^{2}$ inverse theorem gives
$\phi({\textbf{x}}^{\prime})\in{\mathbb{F}}_{p}$ such that

$\displaystyle\left\|f\right\|_{U^{2}({\textbf{v}})}^{4}=\operatorname*{\mathbb{E}}_{{\textbf{x}}^{\prime}\in
H}\left\|f_{{\textbf{x}}^{\prime}}\right\|^{4}_{U^{2}}\leq\operatorname*{\mathbb{E}}_{{\textbf{x}}^{\prime}\in
H}\mathopen{}\left|\operatorname*{\mathbb{E}}_{n}f({\textbf{x}}^{\prime}+{\textbf{v}}n)e_{p}(\phi({\textbf{x}}^{\prime})n)\mathclose{}\right|.$

Picking a phase function $\psi:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ and
$|\lambda|=1$ satisfying

$\displaystyle\operatorname*{\mathbb{E}}_{n}f({\textbf{x}}^{\prime}+{\textbf{v}}n)\cdot\lambda
e_{p}(\phi({\textbf{x}}^{\prime})n+\psi({\textbf{x}}^{\prime}))=\mathopen{}\left|\operatorname*{\mathbb{E}}_{n}f({\textbf{x}}^{\prime}+{\textbf{v}}n)e_{p}(\phi({\textbf{x}}^{\prime})n)\mathclose{}\right|,$

we get

$\displaystyle\left\|f\right\|_{U^{2}({\textbf{v}})}^{4}\leq\operatorname*{\mathbb{E}}_{{\textbf{x}}^{\prime}\in
H}\operatorname*{\mathbb{E}}_{n}f({\textbf{x}}^{\prime}+{\textbf{v}}n)\cdot\lambda
e_{p}(\phi({\textbf{x}}^{\prime})n+\psi({\textbf{x}}^{\prime})).$

The first statement follows upon setting
$\chi({\textbf{x}})=\chi({\textbf{x}}^{\prime}+{\textbf{v}}n)=\lambda
e_{p}(\phi({\textbf{x}}^{\prime})n+\psi({\textbf{x}}^{\prime}))$, and the
second follows from the observation that

$\displaystyle\operatorname*{\mathbb{E}}(f\cdot\chi|{\textbf{v}})({\textbf{x}})$
$\displaystyle=\operatorname*{\mathbb{E}}_{m}f({\textbf{x}}+{\textbf{v}}m)\chi({\textbf{x}}+{\textbf{v}}m)=\operatorname*{\mathbb{E}}_{m}f({\textbf{x}}^{\prime}+{\textbf{v}}(n+m))\chi({\textbf{x}}^{\prime}+{\textbf{v}}(n+m))$
$\displaystyle=\operatorname*{\mathbb{E}}_{m}f({\textbf{x}}^{\prime}+{\textbf{v}}m)\chi({\textbf{x}}^{\prime}+{\textbf{v}}m)=\mathopen{}\left|\operatorname*{\mathbb{E}}_{m}f({\textbf{x}}^{\prime}+{\textbf{v}}m)e_{p}(\phi({\textbf{x}}^{\prime})m)\mathclose{}\right|\geq
0.$

∎

Theorem 1.2 corresponds to the $m=\ell$ case of the following result.

######  Proposition 6.2.

Let $d,D,\ell\in{\mathbb{N}}$, $m\in[0,\ell]$,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{D}$ be nonzero
vectors, and $p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ be linearly independent
polynomials of degrees at most $d$ with zero constant terms. Then

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))-\operatorname*{\mathbb{E}}_{{\textbf{x}}}f_{0}({\textbf{x}})\prod_{j\in[\ell]}\operatorname*{\mathbb{E}}(f_{j}|{\textbf{v}}_{j})({\textbf{x}})\mathclose{}\right|\ll_{d,\ell}p^{-\Omega_{d,\ell}(1)}$

holds for all 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ such that
$f_{j}\in{\mathcal{E}}({\textbf{v}}_{j})$ for all indices $j\in[m+1,\ell]$.

Equivalently, we want to show that the counting operator

$\displaystyle\Lambda(f_{0},\ldots,f_{\ell})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))$

is controlled by the norms
$\left\|f_{1}\right\|_{U^{1}({\textbf{v}}_{1})},\ldots,\left\|f_{\ell}\right\|_{U^{1}({\textbf{v}}_{\ell})}$
with power-saving error terms.

For the rest of this section, we let

(59)
$\displaystyle\tilde{f}_{m}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
m\end{subarray}}f_{0}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n)-{\textbf{v}}_{m}p_{m}(n)),$

where we recall that ${\textbf{v}}_{0}=\mathbf{0}$ and $p_{i}(0)=0$.
Proposition 6.2 will follow from the following result.

######  Proposition 6.3 (Degree lowering).

Let $d,D,\ell\in{\mathbb{N}}$, $m\in[\ell]$,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{D}$ be nonzero
vectors, and $p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ be linearly independent
polynomials of degrees at most $d$ with zero constant terms. Let
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ be 1-bounded
functions such that $f_{j}\in{\mathcal{E}}({\textbf{v}}_{j})$ for all indices
$j\in[m+1,\ell]$, and define $\tilde{f}_{m}$ as in (59). For each $s\geq 2$,
the following holds: if $\delta\gg_{d,\ell,s}p^{-\Omega_{d,\ell,s}(1)}$, then

$\displaystyle\left\|\tilde{f}_{m}\right\|_{U^{s}({\textbf{v}}_{m})}\geq\delta\quad\Rightarrow\quad\left\|\tilde{f}_{m}\right\|_{U^{s-1}({\textbf{v}}_{m})}\gg_{d,\ell,s}\delta^{O_{d,\ell,s}(1)}.$

We first show how Proposition 6.2 for $(m,\ell)$ can be deduced from
Proposition 6.3 for $(m,\ell)$ and Proposition 6.2 for $(m-1,\ell-1)$ (or
$(m-1,\ell)$). Later, we will derive Proposition 6.3 for $(m,\ell)$ by
invoking Proposition 6.2 for $(m-1,\ell)$.

###### Proof of Proposition 6.2 using Proposition 6.3.

Let $\ell\in{\mathbb{N}}$. We split into two cases: $m=0$ and $m>0$.

The case $m=0$.

Suppose first that $m=0$. Then $f_{1},\ldots,f_{\ell}$ are all eigenfunctions
of respective vectors, and so we have

$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))=\operatorname*{\mathbb{E}}_{{\textbf{x}}}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}f_{j}({\textbf{x}})\cdot\operatorname*{\mathbb{E}}_{n}e_{p}\mathopen{}\left(\sum_{j\in[\ell]}\phi_{j}({\textbf{x}})p_{j}(n)\mathclose{}\right),$

where $\phi_{j}$ is the eigenvalue of $f_{j}$. Since the polynomials
$p_{1},\ldots,p_{\ell}$ are linearly independent, we get from the Weil
estimates (see e.g. [18, Theorem 3.2]) that

$\displaystyle\operatorname*{\mathbb{E}}_{n}e_{p}\mathopen{}\left(\sum_{j\in[\ell]}\phi_{j}({\textbf{x}})p_{j}(n)\mathclose{}\right)=1_{\phi_{1}({\textbf{x}})=\cdots=\phi_{\ell}({\textbf{x}})=0}+O_{d}(p^{-1/2}).$

Hence

$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))=\operatorname*{\mathbb{E}}_{{\textbf{x}}}f_{0}({\textbf{x}})\cdot\prod_{j\in[\ell]}(f_{j}({\textbf{x}})\cdot
1_{\phi_{j}({\textbf{x}})=0})+O_{d}(p^{-1/2}),$

and the result follows from (58).

The case $m>0$

We move on to the case $m>0$, i.e. when $f_{m+1},\ldots,f_{\ell}$ are all
eigenfunctions of the respective vectors. Our first goal is to show that under
this assumption, we have

(60)
$\displaystyle|\Lambda(f_{0},\ldots,f_{\ell})|^{O(1)}\ll\min_{j\in[\ell]\setminus\\{m\\}}\left\|f_{j}\right\|_{U^{1}({\textbf{v}}_{j})}+p^{-1}.$

Here and for the rest of the proof, we let all quantities depend on $d$ and
$\ell$. Let $\delta=|\Lambda(f_{0},\ldots,f_{\ell})|$, and suppose that
$\delta\gg p^{c}$ for a sufficiently small $c>0$ \- otherwise the result
follows trivially. By Lemma A.1, we have

$\Lambda(f_{0},\ldots,f_{m-1},\tilde{f}_{m},f_{m+1},\ldots,f_{\ell})\geq\delta^{2},$

and hence by Theorem 1.3, there exists $s\in{\mathbb{N}}$ such that
$\left\|\tilde{f}_{m}\right\|_{U^{s}({\textbf{v}}_{m})}\gg\delta^{O(1)}$. By
an iterated application of Proposition 6.3, we have
$\left\|\tilde{f}_{m}\right\|_{U^{1}({\textbf{v}}_{m})}\gg\delta^{O(1)}$, and
hence Lemma A.1(ii) gives

$\Lambda(f_{0},\ldots,f_{m-1},\operatorname*{\mathbb{E}}(\tilde{f}_{m}|{\textbf{v}}_{m}),f_{m+1},\ldots,f_{\ell})\\\
=\operatorname*{\mathbb{E}}_{\textbf{x}}(f_{0}\operatorname*{\mathbb{E}}(\tilde{f}_{m}|{\textbf{v}}_{m}))({\textbf{x}})\prod_{\begin{subarray}{c}j\in[\ell],\\\
j\neq
m\end{subarray}}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\gg\delta^{O(1)}.$

We have reduced to the case $(m-1,\ell-1)$ (or $(m-1,\ell)$ if we think of 1
as the function evaluated at ${\textbf{x}}+{\textbf{v}}_{m}p_{m}(n)$), and so
invoking Proposition 5.4 in this case, we deduce that
$\left\|f_{j}\right\|_{U^{1}({\textbf{v}}_{j})}\gg\delta^{O(1)}$ for
$j\in[\ell]\setminus\\{m\\}$. Hence (60) follows.

It remains to show that the $U^{1}({\textbf{v}}_{m})$ norm of $f_{m}$ also
controls $\Lambda(f_{0},\ldots,f_{\ell})$. Splitting
$f_{j}=\operatorname*{\mathbb{E}}(f_{j}|{\textbf{v}}_{j})+(f_{j}-\operatorname*{\mathbb{E}}(f_{j}|{\textbf{v}}_{j}))$
for $j\in[\ell]\setminus\\{m\\}$ and using (60), we deduce that

$\displaystyle\Lambda(f_{0},\ldots,f_{\ell})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}F({\textbf{x}})f_{m}({\textbf{x}}+{\textbf{v}}_{m}p_{m}(n))+O(p^{-\Omega(1)}),$

where
$F=f_{0}\cdot\prod_{\begin{subarray}{c}j\in[\ell]\setminus\\{m\\}\end{subarray}}\operatorname*{\mathbb{E}}(f_{j}|{\textbf{v}}_{j})$.
Since we are left with a single dimensional progression, the claim then
follows from [24, Theorem 2.1]. ∎

It remains to prove Proposition 6.3 for $(m,\ell)$ assuming Proposition 5.4
for $(m-1,\ell)$.

###### Proof of Proposition 6.3.

We allow all the quantities in the proof to depend on $d,\ell,s$.

Suppose that
$\left\|\tilde{f}_{m}\right\|_{U^{s}({\textbf{v}}_{m})}\geq\delta$ for some
$s\geq 2$ and $\delta>0$. The induction formula for Gowers norms gives

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}\left\|\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}\right\|^{4}_{U^{2}({\textbf{v}}_{m})}\geq\delta^{2^{s}}.$

By Lemma 6.1, there exist eigenfunctions
$\chi_{\underline{h}}\in{\mathcal{E}}({\textbf{v}}_{m})$ with eigenvalues
$\phi_{\underline{h}}$ satisfying

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}\operatorname*{\mathbb{E}}_{\textbf{x}}\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}({\textbf{x}})\chi_{\underline{h}}({\textbf{x}})\geq\delta^{2^{s}};$

our goal is to show that for many ${\underline{h}}$, the eigenfunctions
$\chi_{\underline{h}}$ can be expressed as a product of low-complexity
functions on many cosets of ${\textbf{v}}_{m}$. Lemma 6.1 also gives

$\operatorname*{\mathbb{E}}(\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{\underline{h}}|{\textbf{v}}_{m})\geq
0;$

hence the set

$\displaystyle B=\\{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}:\
\operatorname*{\mathbb{E}}_{\textbf{x}}\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}({\textbf{x}})\chi_{\underline{h}}({\textbf{x}})\geq\delta^{2^{s}}/2\\}$

has size $|B|\gg\delta^{2^{s}}p^{2^{s-2}}$, and the set

$\displaystyle{\mathcal{U}}_{\underline{h}}=\\{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}:\
\operatorname*{\mathbb{E}}(\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{\underline{h}}|{\textbf{v}}_{m})({\textbf{x}})\geq\delta^{2^{s}}/4\\}$

has cardinality $|{\mathcal{U}}_{\underline{h}}|\gg\delta^{2^{s}}p^{D}$ for
each ${\underline{h}}\in B$. The sets ${\mathcal{U}}_{\underline{h}}$ are
${\textbf{v}}_{m}$-invariant, which gives us the lower bound

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}1_{B}({\underline{h}})\operatorname*{\mathbb{E}}_{\textbf{x}}1_{{\mathcal{U}}_{{\underline{h}}}}({\textbf{x}})\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}({\textbf{x}})\chi_{\underline{h}}({\textbf{x}})\gg\delta^{2^{s}}.$

The next step is to pass the multiplicative derivative from $\tilde{f}_{m}$ to
the functions $f_{0},\ldots,f_{\ell}$. This is accomplished using Proposition
A.2 applied to
$u_{\underline{h}}({\textbf{x}})=1_{B}({\underline{h}})1_{{\mathcal{U}}_{{\underline{h}}}}({\textbf{x}})\chi_{\underline{h}}({\textbf{x}})$,
which gives

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime}}({\underline{h}},{\underline{h}}^{\prime})\operatorname*{\mathbb{E}}_{{\textbf{x}},n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
m\end{subarray}}f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\cdot\mathopen{}\left(1_{{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}}\chi_{{\underline{h}},{\underline{h}}^{\prime}}\mathclose{}\right)({\textbf{x}}+{\textbf{v}}_{m}p_{m}(n))\gg\delta^{O(1)}$

with

$\displaystyle B^{\prime}$
$\displaystyle=\\{({\underline{h}},{\underline{h}}^{\prime})\in{\mathbb{F}}_{p}^{2(s-2)}:\;{\underline{h}}^{\underline{\epsilon}}\in
B\;\textrm{for\; all}\;{\underline{\epsilon}}\in\\{0,1\\}^{s-2}\\}$
$\displaystyle{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}$
$\displaystyle=\bigcap_{{\underline{\epsilon}}\in\\{0,1\\}^{s-2}}{\mathcal{U}}_{{\underline{h}}^{\underline{\epsilon}}}$
$\displaystyle f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$
$\displaystyle=\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}})$
$\displaystyle\chi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$
$\displaystyle=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s-2}}{\mathcal{C}}^{|{\underline{\epsilon}}|}\chi_{{\underline{h}}^{\underline{\epsilon}}}({\textbf{x}}-(\underline{1}\cdot{\underline{h}}^{\prime}){\textbf{v}}_{m}).$

We recall that
${\underline{h}}^{\underline{\epsilon}}=(h_{1}^{\varepsilon_{1}},\ldots,h_{s}^{\varepsilon_{s}})$
with $h_{i}^{\varepsilon_{i}}=h_{i}$ if $\varepsilon_{i}=0$ and
$h_{i}^{\varepsilon_{i}}=h_{i}^{\prime}$ otherwise.

Crucially, each counting operator indexed by
${\underline{h}},{\underline{h}}^{\prime}$ corresponds to the case
$(m-1,\ell)$ of Proposition 5.4, as the function
$1_{{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}}\chi_{{\underline{h}},{\underline{h}}^{\prime}}$
evaluated at ${\textbf{x}}+{\textbf{v}}_{m}p_{m}(n)$ is an eigenfunction of
${\textbf{v}}_{m}$. Applying the case $(m-1,\ell)$ of Proposition 5.4, we
therefore get

(61)
$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime}}({\underline{h}},{\underline{h}}^{\prime})\left\|1_{{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}}\chi_{{\underline{h}},{\underline{h}}^{\prime}}\right\|_{U^{1}({\textbf{v}}_{m})}^{2}\gg\delta^{O(1)}.$

We define

$\displaystyle\phi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=\sum_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}(-1)^{|{\underline{\epsilon}}|}\phi_{{\underline{h}}^{\underline{\epsilon}}}({\textbf{x}}),$

so that

(62)
$\displaystyle\chi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{m}n)=e_{p}(\phi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})n)\chi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}).$

Since ${\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}$ is
${\textbf{v}}_{m}$-invariant, we deduce from (58), (61) and (62) that

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime}}({\underline{h}},{\underline{h}}^{\prime})\operatorname*{\mathbb{E}}_{\textbf{x}}1_{{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}}({\textbf{x}})1_{\phi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=0}\gg\delta^{O(1)}.$

We have thus shown that for many differences
${\underline{h}},{\underline{h}}^{\prime}$ and many points x, the value
$\phi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})$ is 0. We will
use this observation to detect some low-complexity structure on the phases
$\phi_{\underline{h}}.$ First, we apply the pigeonhole principle to find
${\underline{h}}^{\prime}\in B$ for which

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime}}({\underline{h}},{\underline{h}}^{\prime})\operatorname*{\mathbb{E}}_{\textbf{x}}1_{{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}}({\textbf{x}})1_{\phi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=0}\gg\delta^{O(1)}.$

Letting

$\displaystyle B^{\prime\prime}$
$\displaystyle=\\{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}:\
({\underline{h}},{\underline{h}}^{\prime})\in B^{\prime}\\},$
$\displaystyle{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}$
$\displaystyle=\\{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}:\
{\textbf{x}}\in{\mathcal{U}}^{\prime}_{{\underline{h}},{\underline{h}}^{\prime}}\;\textrm{and}\;\phi_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=0\\},$

we deduce that

(63)
$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime\prime}}({\underline{h}})\operatorname*{\mathbb{E}}_{\textbf{x}}1_{{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}}({\textbf{x}})\gg\delta^{O(1)}.$

Unsurprisingly, the sets ${\mathcal{U}}^{\prime\prime}_{\underline{h}}$ are
${\textbf{v}}_{m}$-invariant.

Next, we note that for each ${\underline{h}}\in B^{\prime\prime}$ and
${\textbf{x}}\in{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}$, we can write
$\phi_{{\underline{h}}}({\textbf{x}})=\sum_{j=1}^{s-2}\phi_{j{\underline{h}}}({\textbf{x}})$,
where $\phi_{j{\underline{h}}}$ is defined by

$\displaystyle\phi_{j{\underline{h}}}({\textbf{x}})=\begin{cases}\sum_{\begin{subarray}{c}{\underline{\epsilon}}\in\\{0,1\\}^{s-2},\varepsilon_{j}=1,\\\
\varepsilon_{1}=\ldots=\varepsilon_{j-1}=0\end{subarray}}(-1)^{|{\underline{\epsilon}}|+1}\phi_{{\underline{h}}^{\underline{\epsilon}}}({\textbf{x}}),\;&{\underline{h}}\in
B^{\prime\prime},{\textbf{x}}\in{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}\\\
0,\;&\textrm{otherwise}.\end{cases}$

It follows that for ${\underline{h}}\in B^{\prime\prime}$ and
${\textbf{x}}\in{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}$, we have the
identity

(64)
$\displaystyle\chi_{\underline{h}}({\textbf{x}}+{\textbf{v}}_{m}n)=\prod_{j=1}^{s-2}e_{p}(\phi_{j{\underline{h}}}({\textbf{x}})n)\cdot\chi_{\underline{h}}({\textbf{x}}).$

Crucially, the sequence ${\underline{h}}\mapsto\phi_{j{\underline{h}}}$ does
not depend on $h_{j}$.

Bringing together the inclusions $B^{\prime\prime}\subset B$ and
${\mathcal{U}}^{\prime\prime}_{\underline{h}}\subset{\mathcal{U}}_{\underline{h}}$
for ${\underline{h}}\in B^{\prime\prime}$, the inequality (63) and the lower
bound
$\operatorname*{\mathbb{E}}(\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{\underline{h}}|{\textbf{v}}_{m})({\textbf{x}})\gg\delta^{O(1)}$
for ${\underline{h}}\in B$ and ${\textbf{x}}\in{\mathcal{U}}_{\underline{h}}$,
we obtain the inequality

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime\prime}}({\underline{h}})\operatorname*{\mathbb{E}}_{\textbf{x}}1_{{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}}({\textbf{x}})\operatorname*{\mathbb{E}}(\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{\underline{h}}|{\textbf{v}}_{m})({\textbf{x}})\gg\delta^{O(1)}.$

Combined with the identity (64), it implies that

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s-2}}1_{B^{\prime\prime}}({\underline{h}})\operatorname*{\mathbb{E}}_{{\textbf{x}}}1_{{\mathcal{U}}^{\prime\prime}_{{\underline{h}}}}({\textbf{x}})\chi_{\underline{h}}({\textbf{x}})\operatorname*{\mathbb{E}}_{n}\Delta_{s-2,{\textbf{v}}_{m};{\underline{h}}}\tilde{f}_{m}({\textbf{x}}+{\textbf{v}}_{m}n)e_{p}\mathopen{}\left(\sum_{j=1}^{s-2}\phi_{j{\underline{h}}}({\textbf{x}})n\mathclose{}\right)\gg\delta^{O(1)}.$

An application of the Cauchy-Schwarz inequality and the
${\textbf{v}}_{m}$-invariance of $\phi_{j{\underline{h}}}$ give

$\displaystyle\operatorname*{\mathbb{E}}_{({\underline{h}},h_{s-1})\in{\mathbb{F}}_{p}^{s-1}}\operatorname*{\mathbb{E}}_{{\textbf{x}}}\Delta_{s-1,{\textbf{v}}_{m};({\underline{h}},h_{s-1})}\tilde{f}_{m}({\textbf{x}})e_{p}\mathopen{}\left(-\sum_{j=1}^{s-2}\phi_{j{\underline{h}}}({\textbf{x}})h_{s-1}\mathclose{}\right)\gg\delta^{O(1)}.$

The result follows from Lemma A.4.

∎

##  Appendix A Standard technical lemmas

We gather here standard technical lemmas that we need for various arguments.
Throughout this section, we set

$\displaystyle\Lambda(f_{0},\ldots,f_{\ell})$
$\displaystyle=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\prod_{j\in[\ell]}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))$
$\displaystyle\textrm{and}\quad\quad\tilde{f}_{m}({\textbf{x}})$
$\displaystyle=\operatorname*{\mathbb{E}}_{n}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
m\end{subarray}}f_{0}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n)-{\textbf{v}}_{m}p_{m}(n)).$

for some vectors
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{d}$, polynomials
$p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ and 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$.

The following trick has become pretty standard in degree lowering and norm
smoothing arguments.

######  Lemma A.1.

Suppose that
$\operatorname*{\mathbb{E}}_{\textbf{x}}\tilde{f}_{m}({\textbf{x}})g({\textbf{x}})\geq\delta$
for a 1-bounded function $g:{\mathbb{F}}_{p}\to{\mathbb{C}}$. Then

$\displaystyle\Lambda(f_{0},\ldots,f_{m-1},\overline{g},f_{m+1},\ldots,f_{\ell})\geq\delta.$

In particular,

  1. (i)

if $|\Lambda(f_{0},\ldots,f_{\ell})|\geq\delta$, then
$\Lambda(f_{0},\ldots,f_{m-1},\tilde{f}_{m},f_{m+1},\ldots,f_{\ell})\geq\delta^{2}$;

  2. (ii)

if $\left\|\tilde{f}_{m}\right\|_{U^{1}({\textbf{v}}_{m})}\geq\delta$, then
$\Lambda(f_{0},\ldots,f_{m-1},\operatorname*{\mathbb{E}}(\tilde{f}_{m}|{\textbf{v}}_{m}),f_{m+1},\ldots,f_{\ell})\geq\delta^{2}$;

  3. (iii)

if $\left\|\tilde{f}_{m}\right\|_{U^{s}({\textbf{v}}_{m})}\geq\delta$, then
$\Lambda(f_{0},\ldots,f_{m-1},\overline{{\mathcal{D}}_{s,{\textbf{v}}_{m}}\tilde{f}_{m}},f_{m+1},\ldots,f_{\ell})\geq\delta^{2^{s}}$;

###### Proof.

Since
$\operatorname*{\mathbb{E}}_{\textbf{x}}\tilde{f}_{m}({\textbf{x}})g({\textbf{x}})$
is real by assumption, it equals its complex conjugate
$\operatorname*{\mathbb{E}}_{\textbf{x}}\overline{\tilde{f}_{m}({\textbf{x}})g({\textbf{x}})}$.
The first claim is then a direct consequence of the identity

$\displaystyle\operatorname*{\mathbb{E}}_{\textbf{x}}\overline{\tilde{f}_{m}({\textbf{x}})g({\textbf{x}})}=\Lambda(f_{0},\ldots,f_{m-1},\overline{g},f_{m+1},\ldots,f_{\ell}),$

which follows from expanding the definition of $\tilde{f}_{m}$ and making the
change of variables
${\textbf{x}}\mapsto{\textbf{x}}+{\textbf{v}}_{m}p_{m}(n)$. The claim (i)
follows from

$\displaystyle|\Lambda(f_{0},\ldots,f_{\ell})|^{2}=|\operatorname*{\mathbb{E}}_{\textbf{x}}f_{m}({\textbf{x}})\overline{\tilde{f}_{m}({\textbf{x}})}|^{2}\leq\left\|\tilde{f}_{m}\right\|_{2}^{2}=\operatorname*{\mathbb{E}}_{\textbf{x}}\tilde{f}_{m}({\textbf{x}})\overline{\tilde{f}_{m}({\textbf{x}})}$

upon taking $g=\overline{\tilde{f}_{m}({\textbf{x}})}$.

The claims (ii) and (iii) follow from the weak inverse theorems for
$U^{1}({\textbf{v}}_{m})$ and $U^{s}({\textbf{v}}_{m})$ upon taking $g$ to be
$\overline{\operatorname*{\mathbb{E}}(\tilde{f}_{m}|{\textbf{v}}_{m})}$ and
${\mathcal{D}}_{s,{\textbf{v}}_{m}}\tilde{f}_{m}$ respectively. ∎

The next result allows us to pass differences from the structured function
$\tilde{f}_{m}$ to its component functions. Its proof, based on $s$
applications of the Cauchy-Schwarz inequality followed by the change of
variables
${\textbf{x}}\mapsto{\textbf{x}}-(h_{1}^{\prime}{\textbf{b}}_{1}+\cdots+h^{\prime}_{s}{\textbf{b}}_{s})$,
is pretty standard and follows the same strategy as the proofs of [7,
Proposition 4.3] or [29, Lemma 6.3].

######  Lemma A.2 (Dual-difference interchange).

Let $s,D\in{\mathbb{N}}$,
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}\in{\mathbb{Z}}^{D}$ and for each
${\underline{h}}\in F_{p}^{s}$, let
$u_{\underline{h}}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ be 1-bounded. If

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{\textbf{x}}\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}}\tilde{f}_{m}({\textbf{x}})u_{\underline{h}}({\textbf{x}})\geq\delta$

for some $\delta>0$, then

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{\textbf{x}}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
m\end{subarray}}\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{m}p_{m}(n))\geq\delta^{2^{s}},$

where

$\displaystyle
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}u_{{\underline{h}}^{{\underline{\epsilon}}}}({\textbf{x}}-(h_{1}^{\prime}{\textbf{b}}_{1}+\cdots+h^{\prime}_{s}{\textbf{b}}_{s})).$

In applications, we will mostly need the following corollary of Lemma A.2,
which combines Lemma A.2 with the inductive formula for box norms and weak
inverse theorem for Gowers norms.

######  Corollary A.3.

Let $s,s^{\prime},D\in{\mathbb{N}}$,
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\textbf{b}}\in{\mathbb{Z}}^{D}$ and
suppose that

$\displaystyle\left\|\tilde{f}_{m}\right\|_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\textbf{b}}^{\times
s^{\prime}}}\geq\delta$

for some $\delta>0$. Then

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{\textbf{x}}\prod_{\begin{subarray}{c}j\in[0,\ell],\\\
j\neq
m\end{subarray}}\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{m}p_{m}(n))\geq\delta^{2^{2s+s^{\prime}}},$

where

  1. (i)

if $s^{\prime}=1$, then $u_{{\underline{h}},{\underline{h}}^{\prime}}$ is
b-invariant;

  2. (ii)

if $s^{\prime}\geq 1$, then $u_{{\underline{h}},{\underline{h}}^{\prime}}$ is
a product of $2^{s}$ elements of ${\mathfrak{D}}_{s^{\prime}}({\textbf{b}})$.

###### Proof.

From the inductive formula for box norms and the weak inverse theorem for
$U^{s^{\prime}}({\textbf{b}})$, we have

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{\textbf{x}}\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}}\tilde{f}_{m}({\textbf{x}})u_{\underline{h}}({\textbf{x}})\geq\delta^{2^{s+s^{\prime}}},$

where $u_{\underline{h}}\in{\mathfrak{D}}_{s^{\prime}}({\textbf{b}})$. If
$s^{\prime}=1$, then $u_{\underline{h}}$ is additionally b-invariant. The
claim then follows from Lemma A.2 and the observation that
$u_{{\underline{h}},{\underline{h}}^{\prime}}$ is a product of $2^{s}$
elements of ${\mathfrak{D}}_{s^{\prime}}({\textbf{b}})$ (and b-invariant for
$s^{\prime}=1$). ∎

The last lemma is a standard variant of the Gowers-Cauchy-Schwarz inequality
that allows us to remove low complexity functions, and it can be proved the
same way as [7, Lemma 3.4] or [29, Lemma 6.4].

######  Lemma A.4 (Removing low-complexity functions).

Let $D,s\in{\mathbb{N}}$, ${\textbf{v}}\in{\mathbb{Z}}^{D}$ be a vector
$f:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ be a 1-bounded function. For $j\in[s]$
and ${\underline{h}}\in{\mathbb{F}}_{p}^{s}$, let
$g_{j{\underline{h}}}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ be 1-bounded
functions such that the sequence of functions ${\underline{h}}\mapsto
g_{j{\underline{h}}}$ does not depend on the variable $h_{j}$. Then

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{{\textbf{x}}}\Delta_{s,{\textbf{v}};{\underline{h}}}f({\textbf{x}})\cdot\prod_{j=1}^{s}g_{j{\underline{h}}}({\textbf{x}})\mathclose{}\right|\leq\left\|f\right\|_{U^{s}({\textbf{v}})}.$

## References

  * [1] E. Ackelsberg and V. Bergelson.  Polynomial patterns in subsets of large finite fields of low characteristic.  Preprint 2023, arXiv:2303.00925.
  * [2] V. Bergelson and A. Best.  On the polynomial Szemerédi theorem in finite commutative rings.  Preprint 2023, arXiv:2303.17928.
  * [3] J. Bourgain and M.-C. Chang.  Nonlinear Roth type theorems in finite fields.  Israel J. Math., 221:853–867, 2017.
  * [4] D. Dong, X. Li, and W. Sawin.  Improved estimates for polynomial Roth type theorems in finite fields.  J. Anal. Math., 141:689–705, 2020.
  * [5] S. Donoso, A. Ferré-Moragues, A. Koutsogiannis, and W. Sun.  Decomposition of multicorrelation sequences and joint ergodicity.  Preprint 2021, arXiv:2106.01058.
  * [6] S. Donoso, A. Koutsogiannis, and W. Sun.  Seminorms for multiple averages along polynomials and applications to joint ergodicity.  J. Anal. Math., 146(1):1–64, 2022.
  * [7] N. Frantzikinakis.  Joint ergodicity of sequences.  Preprint 2021, arXiv:2102.09967, to appear in Adv. Math.
  * [8] N. Frantzikinakis.  A multidimensional Szemerédi theorem for Hardy sequences of different growth.  Trans. Amer. Math. Soc., 367:5653–5692, 2012.
  * [9] N. Frantzikinakis and B. Host.  Weighted multiple ergodic averages and correlation sequences.  Ergodic Theory Dynam. Systems, 38(1):81–142, 2018.
  * [10] N. Frantzikinakis and B. Kuca.  Joint ergodicity for commuting transformations and applications to polynomial sequences.  Preprint 2022, arXiv:2207.12288.
  * [11] N. Frantzikinakis and B. Kuca.  Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates.  Preprint 2022, arXiv:2209.11033, to appear in Ergodic Theory Dynam. Systems.
  * [12] B. Green.  Montreal lecture notes on quadratic Fourier analysis, 2007.  arXiv:0604089.
  * [13] B. Green and T. Tao.  An inverse theorem for the Gowers $U^{3}(G)$ norm.  Proc. Edinb. Math. Soc., 51(1):73–153, 2008.
  * [14] B. Green, T. Tao, and T. Ziegler.  An inverse theorem for the Gowers $U^{4}$ norm.  Glasg. Math. J., 53(1):1–50, 2011.
  * [15] B. Green, T. Tao, and T. Ziegler.  An inverse theorem for the Gowers $U^{s+1}[N]$-norm.  Ann. of Math., 176(2):1231–1372, 2012.
  * [16] R. Han, M. T. Lacey, and F. Yang.  A polynomial Roth theorem for corners in finite fields.  Mathematika, 67(4):885–896, 2021.
  * [17] B. Host.  Ergodic seminorms for commuting transformations and applications.  Studia Math., 195:31–49, 2009.
  * [18] E. Kowalski.  Exponential sums over finite fields: elementary methods.  https://people.math.ethz.ch/ kowalski/exp-sums.pdf.
  * [19] B. Kuca.  Multidimensional polynomial Szemerédi theorem in finite fields for distinct-degree polynomials.  Preprint 2021, arXiv:2104.07339, to appear in Israel J. Math.
  * [20] B. Kuca.  Further bounds in the polynomial Szemerédi theorem over finite fields.  Acta Arith., 198:77–108, 2021.
  * [21] J. Leng.  A quantitative bound for Szemerédi’s theorem for a complexity one polynomial progression over ${\mathbb{Z}}/N{\mathbb{Z}}$.  Preprint 2022, arXiv:2205.05540.
  * [22] F. Manners.  Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups.  Preprint 2018, arXiv:1811.00718.
  * [23] S. Peluse.  Three-term polynomial progressions in subsets of finite fields.  Israel J. Math., 228:379–405, 2018.
  * [24] S. Peluse.  On the polynomial Szemerédi theorem in finite fields.  Duke Math. J., 168(5):749–774, 2019.
  * [25] S. Peluse.  Bounds for sets with no polynomial progressions.  Forum Math. Pi, 8, 2020.
  * [26] S. Peluse and S. Prendiville.  Quantitative bounds in the non-linear Roth theorem.  Preprint 2019, arXiv:1903.02592.
  * [27] S. Peluse and S. Prendiville.  A polylogarithmic bound in the nonlinear Roth theorem.  Int. Math. Res. Not., 2020.  rnaa261.
  * [28] S. Prendiville.  Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case.  Discrete Anal., 5, 2017.
  * [29] S. Prendiville.  The inverse theorem for the nonlinear Roth configuration: an exposition, 2020.  Preprint 2020, arXiv:2003.04121.
  * [30] T. Tao and V. Vu.  Additive Combinatorics.  Cambridge Studies in Advanced Mathematics. Cambridge U. P., 2006.
  * [31] T. Tao and T. Ziegler.  Concatenation theorems for anti-Gowers uniform functions and Host- Kra characteristic factors.  Discrete Anal., 13, 2016.
  * [32] T. Tao and T. Ziegler.  Polynomial patterns in the primes.  Forum Math. Pi, 6, 2018.


Analysis of radiation damage in silicon charge-coupled devices used for dark
matter searches.

Steven J. Lee1$\star$ on behalf of DAMIC-M collaboration

1 Universität Zürich Physik Institut, Zürich, Switzerland

<EMAIL_ADDRESS>
14th International Conference on Identification of Dark Matter
Vienna, Austria, 18-22 July 2022 10.21468/SciPostPhysProc.?

## Abstract

Nuclear recoils in crystal detectors generate radiation damage in the form of
crystal defects that can be measured in scientific-grade CCDs as local hot
spots of leakage current stimulated by temperature increases in the devices.
In this proceeding, we use a neutron source to generate defects in DAMIC-M
CCDs, and using increases in leakage current at different temperatures, we
demonstrate a procedure for identifying crystal defects in the CCDs of the
DAMIC-M experiment. This is the first time that individual defects generated
from nuclear recoils have been studied. This technique could be used to
distinguish nuclear recoils from electron recoils in some energy ranges, which
would improve the ability of CCD detectors to search for weakly interacting
dark matter.

##  1 Introduction

The DAMIC (Dark Matter in CCDs) collaboration has been searching for dark
matter using charge-coupled devices (CCDs)[1]. The CCDs are solid state
devices that use the silicon bulk as the ionizing medium. The CCDs are unique
compared to the dominant imaging CMOS technology in a way that the CCDs
utilize poly-silicon gate structures and field stops to isolate charge packets
from adjacent pixels. The advantage of using CCDs as particle detector is that
the coupled pixels in the CCDs share a common channel throughout the device
and transferring charge packets from one pixel to another has a very high
charge transfer efficiency. In addition, even for high substrate voltage the
CCD produces very little leakage current making CCDs ideal as high sensitivity
particle detectors.

The DAMIC CCDs have a charge transfer inefficiency less than $10^{-4}$% and
leakage current of approximately
$2\text{\,}\mathrm{e}$-$\text{\,}{\mathrm{mm}}^{-1}\text{\,}{\mathrm{d}}^{-1}$[2].
A modern intervention of DAMIC-CCDs also include skipper amplifiers which
utilize floating readout transistor structures allowing non-destructive
sampling of charge packets on the readout. By increasing the number of
sampling using the skipper amplifier, the DAMIC-M CCDs achieve a readout noise
as low as $\sigma_{RMS}=$$0.05\text{\,}\mathrm{e}$-[3].

In addition to ionizing signals, we can also observe non-ionizing events
within the CCDs that result in observable lattice damage. In this proceeding,
we report on the observation of radiation damage from the defects caused by
single nuclear recoils in DAMIC-M CCDs. The properties of radiation damage
allow us to search for signs of particle-lattice interactions which may be
used in the search for dark matter.

##  2 Lattice defects and dark matter search

When a foreign particle enters a solid state structure, it loses its energy
primarily by Ionizing Energy Loss (IEL) and Non-Ionizing Energy Loss (NIEL)
until the particle escapes the solid completely or stops. The total energy
loss of the particle is equal to the stopping power of the particle[4, 6, 5].

$S=S_{e}+S_{n}+S_{nr}$ (1)

In Equation 1, S is the stopping power, $S_{e}$ is the electronic stopping
power equivalent to IEL, $S_{n}$ is the nuclear stopping power equivalent to
NIEL, $S_{nr}$ is the high relativistic term. It is important to note that the
NIEL or $S_{n}$ component damages the lattice structure of the solids[5].

###  2.1 Defect creation

In the first few picoseconds after a foreign particle collides with a lattice
atom several things happen; the first lattice atom becomes dislodged from its
original location (primary knock-on atom, PKA) and knocks out secondary and
tertiary atoms forming clusters of defects. Figure 1(left) demonstrates
interaction in a diagram. An intrinsic silicon crystal has tetrahedral (4
prong) bonds to each other, and as soon as one of the bonds is broken, new
bonds are made to the neighbouring atoms. Some of the broken bonds are
replaced by Si-Si double or triple bonds, some bonds are made with common
impurities such as oxygen, hydrogen, carbon and etc.

|
---|---
Figure 1: Types of defects produced in radiation damage[4].

Figure 1(right) outlines the types of defects we can expect from radiation
damage. After the first few picoseconds, the lattice is deformed due to these
bonds and stabilizes. The stabilized deformations in the lattice structure are
called crystal defects. Some semiconductor detectors are intentionally
deformed by ion implantation for in-situ or additional doping.

###  2.2 Defect identification

Crystal defects also have a macroscopic electrical property due to the change
in the local band-gap resulting from the different allowed energy states from
the new lattice bonds. As a result the defective regions can introduce a
higher or lower leakage current than the rest of the silicon.

The bulk region of the DAMIC-M CCDs are n type and doped with Phosphorous (P).
Figure 2 (shows) a sketch of the bandgap structure between the valence band
($E_{V}$) and conduction band ($E_{C}$) of silicon. During device production,
Boron (B) is typically used for doping silicon positively and Phosphorous is
typically used for doping silicon negatively, Vacancy defects (V) can mix with
themselves ($V_{2}$), or with Oxygen (O), Hydrogen (H), Carbon (C), and
likewise for Interstitial defects (I). In addition, there are some defects
corresponding to a specific bandgap deformation but not yet identified (X).
The type of defects produced following a radiation damage is almost entirely
stochastic[5]. However all defects have observable macroscopic properties.

|
---|---
Figure 2: Defect species in band gap structure[7] and their thermal
responses[6].

Different defect species respond differently to environmental changes such as
temperature, bias voltage, and stress. One of the easiest characteristics to
test is temperature. Thermally stimulated currents (TSC) is a technique to
measure the different temperature response due to different defect species[6].

Figure 2 (right) shows an example of the temperature dependence of a TSC
signals for four different irradiated samples. The plot shows the abnormal
current generation at different temperatures due to the different defect
species created.

##  3 Neutron Irradiation Campaign

Typically in high-energy particle physics applications, where the goal is to
study performance of irradiated silicon devices, very high energies and
fluences are utilized. However the advantages of using the DAMIC-M CCDs are
their low energy threshold and the ability to discern individual events. These
CCDs can be used to study defect production at the single-event scale.

In the first quarter of 2022 at the University of Washington Center for
Experimental Nuclear Physics and Astrophysics(CENPA) CCD test stand, we
irradiated a 24 megapixel DAMIC-M CCD (UW6418) using an AmBe source which
emits 7400 neutrons per second.

###  3.1 Data collection

All of the data in the experiment were taken in full image mode with
10-minutes exposure followed by 7 minutes of readout time. The CCD temperature
was incremented by $5\text{\,}\mathrm{K}$ ranging from
$120\text{\,}\mathrm{K}$ to $200\text{\,}\mathrm{K}$. For consistency, 8
exposures were taken during and after temperature increase. In between each
data set, the CCD was warmed to room temperature for reconfiguration inside
the test chamber. Table 1 summarizes the 8 data sets of the experiment.

Data set | Type | Source arrangement | Temp. Range(K)
---|---|---|---
Initial test | Permanent defect search | No source | 120 to 200
Fe-55 Calibration | Pixel value-Fe55 x-ray peaks | Fe-55 | 120 to 200
Proof of Concept | Damage intensity test | Unshielded AmBe | 120 to 200
Before Irradiation 1 | Defect search | No source | 120 to 200
Irradiation 1 | Defect production | Unshielded AmBe | 120 to 160
After Irradiation 1 | Defect search | No source | 120 to 200
Irradiation 2 | Defect production | AmBe | 120 to 200
After Irradiation 2 | Defect search | No source | 120 to 200
Table 1: Experiments performed with UW6418 DAMIC-M CCDs at CENPA

The initial test seen in Table 1 was performed without the AmBe source to
identify and eliminate defects already present in the CCD prior to neutron
irradiation. An 55Fe Calibration was then performed to calibrate the raw pixel
values in Arbitrary Digital Units (ADU) to electron-equivalent energy. A Proof
of Concept irradiation was performed with the AmBe source placed less than
$2\text{\,}\mathrm{cm}$ in front of the CCD with minimal lead shielding fir 3
days to confirm the presence of ionization events from the source in the CCD
images. Two other irradiations were also carried out:

  1. 1.

Irradiation 1 (5 days): a longer irradiation with the same source
configuration, which was halted half way after reaching 160 K due to time
constraints.

  2. 2.

Irradiation 2 (6 days): with the AmBe source placed approximately
$11\text{\,}\mathrm{cm}$ in front of the CCD with $8\text{\,}\mathrm{cm}$ of
lead shielding between the CCD and the source.

Data runs without the source (Before Irradiation 1, After Irradiation 1, After
Irradiation 2) were performed after each irradiation to look for newly
produced defects.

###  3.2 Defect analysis

The defect analysis was performed by using data collected from initial test,
before and after irradiations.

From each data set, the images taken at $200\text{\,}\mathrm{K}$ were used to
locate the repeating patterns. All patterns ranging from single pixels up to
vertical saturation lines stretching throughout the CCD were identified as
’tracks’ if the edge of the pattern satisfied Equation 3 in the Appendix.

Only the tracks that appear in more than two images were considered as
defects. A typical physical permanent defect unrelated to the irradiation
generate significantly larger leakage current and saturates into adjacent
pixels. They were easily identified and discarded from the analysis. Most of
the defects produced following the neutron irradiation were typically confined
to one pixel.

Once the defects have been located, the pixel coordinates were used from all
of the data sets. Defects seen from After Irradiation 1 were compared to the
defects seen in Before Irradiation 1 to identify newly produced defects. Same
analysis was performed After Irradiation 2. As seen in Figure 3 the number of
newly produced defects increased from irradiation.

In addition, the originating ionization events leading to the defects were
identified from the correlation of their spatial coordinates in the images.
For ionization events with total energy higher than
$100\text{\,}\mathrm{keV}$ee, nuclear recoils from neutron scattering can be
identified by their topology since they generate symmetrical round clusters,
unlike the extended tracks or “worms” from electron-recoil backgrounds. A
preliminary analysis shows that at least 80% of the nuclear recoils identified
by topology are spatially correlated with a defect appearing after
irradiation. Lower-energy ionization events are also spatially correlated with
defects but, since we cannot tell which events are nuclear recoils, an
efficiency estimate is not possible at this time.

We have also been able to categorize the defects by their thermal responses.
Depending of the type of defects, similar to the TSC analysis, the defects are
stimulated the most at specific temperature. The thermal responses were
categorized if at below $190\text{\,}\mathrm{K}$ the defect satisfies 2.

$z\geq\mu_{z}^{CCD}+1.5\sigma_{z}^{CCD}$ (2)

Here $z$ is the leakage charge from a single (defective) pixel, and
$\mu_{z}^{CCD}$ is the average leakage charge of the whole CCD and
$\sigma_{z}^{CCD}$ is the standard deviation of the leakage charge of the
whole CCD. Using this approach, defects and their response to thermal
stimulation was studied.

##  4 Conclusion

By irradiating a silicon CCD with a neutron AmBe source, we have identified
for the first time individual defects generated by nuclear recoil events. A
preliminary spatial correlation analysis concludes that at least 80% of
recoils with energies >$100\text{\,}\mathrm{keV}$ee produce measurable
defects.

In addition, we have analyzed the thermal stimulation properties of the
defects which can be seen in Figure 3.

Figure 3: Number of defects identified in between neutron irradiations.

Figure 3 shows the number of defects identified before and after each
irradiation. The different colours represent the defects produced prior to and
after each irradiation, blue is after the first proof of concept irradiation,
red is after the second irradiation and green is after third irradiation. In
addition, the defects have the largest gain at outlined cold finger
temperatures.

Future work on radiation defects in CCDs is being studied within the RADAC
(Radiation Damage in CCDs) group, including the potential application of
searching for a daily modulation in the rate of WIMP interactions [8].

The defect analysis is further being developed to search for defects produced
from lower recoil energies to look for a low mass DM search using defects.

## Acknowledgements

#### RD50

The author would like to acknowledge M. Moll, A. Himmerlich, E.C. Rivera A.
Macchiolo, V. Subert and Y. Gurimskaya for providing insight in searching for
radiation damage using TSC and DLTS techniques and providing simulation
support for damage spectrum.

#### Theorists

The author would like to acknowledge K. Nordlund, F. Djurabekova, F.
Kadribasic for providing theoretical insight for damage search and the
application of lattice damage in dark matter search.

#### Funding information

The author would like to acknowledge the Swiss National Science Foundation
(SNSF), the graduate campus of University of Zurich and the University of
Zurich Candoc grant (ehemals Forschungskredit) for enabling this research.

##  Appendix A Edge gradient search

$\left.\frac{dz_{n,m}}{dx_{i}}\right\rvert_{i=n-2}^{i=n+2}\text{ or
}\left.\frac{dz_{n,m}}{dy_{j}}\right\rvert_{j=m-2}^{j=m+2}\geq\mu_{z}^{25adj.pix}+1.5\sigma_{z}^{25adj.pix}$
(3)

In Equation 3, $z_{n,m}$ is pixel value (leakage charge collected from an
individual pixel) of the pixel index n and m, x is pixel location in
horizontal axis of the image, y is pixel number in vertical axis of the image.

$\mu_{z}^{25adj.pix}=\frac{\sum_{n-2,m-2}^{n+2,m+2}z_{i,j}}{25}$ (4)

$\mu_{z}^{25adj.pix}$ is the inclusive average pixel values of 25 surrounding
pixels from $i=n-2$ to $i=n+2$ and $j=m-2$ to $j=m+2$.

$\sigma_{z}^{25adj.pix}=\sqrt{\frac{\sum_{n-2,m-2}^{n+2,m+2}(z_{i,j}-\mu_{z}^{25adj.pix})^{2}}{25}}$
(5)

Similarly $\sigma_{z}^{25adj.pix}$ is the inclusive standard deviation of the
pixel values of 25 surrounding pixels from $i=n-2$ to $i=n+2$ and $j=m-2$ to
$j=m+2$.

## References

  * [1] A. Aguilar-Arevalo, D. Amidei, X. Bertou, D. Bole, M. Butner, G. Cancelo, A. Castañeda Vásquez, A. E. Chavarria, J. R. T. de Mello Neto and S. Dixon, et al. “Measurement of radioactive contamination in the CCD’s of the DAMIC experiment,” J. Phys. Conf. Ser. 718 (2016) no.4, 042057 doi:10.1088/1742-6596/718/4/042057
  * [2] A. Aguilar-Arevalo et al. [DAMIC], “Constraints on Light Dark Matter Particles Interacting with Electrons from DAMIC at SNOLAB,” Phys. Rev. Lett. 123 (2019) no.18, 181802 doi:10.1103/PhysRevLett.123.181802 [arXiv:1907.12628 [astro-ph.CO]].
  * [3] C. De Dominicis et al. [DAMIC-M], “Simulations and background estimate for the DAMIC-M experiment,” Nuovo Cim. C 45 (2021) no.1, 6 doi:10.1393/ncc/i2022-22006-y
  * [4] K. Nordlund and F. Djurabekova, http://www.acclab.helsinki.fi/~knordlun/pub/Nor13apreprint.pdf “Multiscale modelling of irradiation in nanostructures, J. Comput. Electr. 13, 122 (2014)," invited review paper for special issue on device modelling.
  * [5] Personal correspondence with Kai Nordland within RADAC.
  * [6] Michael Moll, “Displacement Damage in Silicon Detectors for High Energy Physics" IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 65, NO. 8, AUGUST 2018 1561
  * [7] J. Stahl, “Defect Characterisation in High-Purity Silicon after $\gamma$\- and Hadron Irraditation." PhD thesis, University of Hamburg, 2004. DESY-THESIS-2004-028, July 2004 (ZEUS).‘
  * [8] F. Kadribasic, N. Mirabolfathi, K. Nordlund and F. Djurabekova, “Crystal Defects: A Portal To Dark Matter Detection,” [arXiv:2002.03525 [physics.ins-det]].