# An Energy Bound in the Affine Group

Giorgis Petridis, Oliver Roche-Newton, Misha Rudnev and Audie Warren

###### Abstract

We prove a nontrivial energy bound for a finite set of affine transformations
over a general field and discuss a number of implications. These include new
bounds on growth in the affine group, a quantitative version of a theorem by
Elekes about rich lines in grids. We also give a positive answer to a question
of Yufei Zhao that for a plane point set $P$ for which no line contains a
positive proportion of points from $P$, there may be at most one line, meeting
the set of lines defined by $P$ in at most a constant multiple of $|P|$
points.

##  1 Introduction

Let $\mathbb{F}$ be a field with multiplicative group $\mathbb{F}^{*}$. We are
particularly interested in the zero characteristic case
$\mathbb{F}=\mathbb{R}$ (or $\mathbb{C}$, there is no difference as far as the
results we present are concerned) or $\mathbb{F}=\mathbb{F}_{p}$, where $p$ is
an odd prime (there is no difference as to our results between
$\mathbb{F}_{p}$ and a general $\mathbb{F}$ with characteristic $p$).

Let $A$ be a finite set of transformations in the affine group
$G=\emph{Aff}(\mathbb{F})=\mathbb{F}\rtimes\mathbb{F}^{*}$. For $g\in G$ we
write $g=(g_{1},g_{2})\in\mathbb{F}^{*}\times\mathbb{F}$ and visualise it as a
point in $\mathbb{F}^{2}$ with the $y$-axis deleted. $G$ has a unipotent
normal subgroup $U=\\{(1,x):x\in\mathbb{F}\\}$ and maximal tori
$T=T(x)=\emph{Stab}(x)$, for the action $g(x)=g_{1}x+g_{2}$ on $\mathbb{F}$.
The tori correspond to finite slope lines through the identity $(1,0)$ in the
coordinate plane. Their left cosets are given by families of parallel lines
with the given slope; their right cosets by families of lines incident to the
deleted $y$-axis at the point where the line corresponding to $T$ itself meets
it. Cosets of $U$ are vertical lines.

Below we use the notation $A^{k}$ for the $k$-fold Cartesian product of $A$
with itself, not to be confused with $A^{-1}$, the set of inverses of elements
of $A$ or the product sets $A^{-1}A,AA$, etc. We define two versions of the
energy of $A$ as

$E(A):=|\\{(g,h,u,v)\in A^{4}:\,g^{-1}\circ h=u^{-1}\circ
v\\}|\,,\;E^{*}(A):=|\\{(g,h,u,v)\in A^{4}:\,g\circ h=u\circ v\\}|\,.$ (1)

These are different quantities. A standard application of the Cauchy-Schwarz
inequality yields the following bounds connecting $E(A)$ and $E^{*}(A)$ with
$A^{-1}A$ and $AA$, respectively.

$E(A)\geq\frac{|A|^{4}}{|A^{-1}A|}\,,\,\,\,\,\,\,\,\,\,E^{*}(A)\geq\frac{|A|^{4}}{|AA|}\,.$
(2)

In fact, Shkredov [17, Section 4] proved the inequality $E^{*}(A)\leq E(A)$.

Also, for a scalar set $S$ we use the standard notations $E^{+}(S)$ (and
respectively $E^{\times}(S)$) for its additive (and respectively
multiplicative) energy.

Note that in principle one cannot expect to have a nontrivial upper bound on
$E(A)$, since $A$ can lie in a coset of an Abelian subgroup. But the paradigm
of the rich theory of growth in groups, developed in the past 15 years, is
that this is the only case when $A$ would not grow by multiplication. The
methods it uses, however, are not only very general, applying to a wide class
of finite non-commutative groups to yield “reasonable” quantitative estimates
on the growth rate, but also technically work with sets, rather than their
energies. Thus but for the case of sufficiently large sets, where one can use
Fourier analysis and representation theory to describe quasirandom group
actions (see e.g. [4] and references therein), the only way to connect this
with the energy of a set of transformations has been via the non-commutative
Balog-Szemerédi-Gowers Theorem.

The first nontrivial instance of a non-commutative group is the affine group.
An element $(g_{1},g_{2})\in G$ can be identified with the line
$y=g_{1}x+g_{2}$. In [15] the following incidence theorem was proved,
establishing an explicit connection between energy in the affine group and
incidence theory.111Here and throughout the paper, for positive values $X$ and
$Y$, the notation $X\gg Y$ and $Y\ll X$ indicates that there is an absolute
constant $C>0$ such that $X\geq CY$.

######  Theorem 1.1.

Let $S,T\subseteq\mathbb{F}$ be finite scalar sets and ${A}$ a finite set of
non-vertical, non-horizontal lines in $\mathbb{F}^{2}$.
If $\mathbb{F}=\mathbb{R}$, then

${I}(S\times
T,{A})\ll|T|^{1/2}|S|^{2/3}E^{1/6}({A})|{A}|^{1/3}+|T|^{1/2}|{A}|\,.$ (3)

If $\mathbb{F}=\mathbb{F}_{p}$, then

${I}(S\times
T,{A})\ll|T|^{1/2}|S|^{5/8}E^{1/8}({A})|{A}|^{1/2}+|T|^{1/2}|{A}|\cdot\sqrt{\max\\{1,|S|^{2}/p\\}}\,.$
(4)

Theorem 1.1 (along with other results) enabled [15, Theorem 7], which is a
somewhat stronger version of the following classical theorem of Elekes [2] on
rich lines in grids.

######  Theorem 1.2 (Elekes).

Let $\alpha\in(0,1)$, $n$ be a positive integer, $A\subset\mathbb{R}$ with
$|A|=n$ and suppose there are $n$ non-horizontal and non-vertical lines
intersecting $A\times A$ in at least $\alpha n$ points. Then either
$\bullet~{}$ at least $\gg\alpha^{C}n$ of these lines are parallel, or
$\bullet~{}$ at least $\gg\alpha^{C}n$ of these lines are incident to a common
point.
Here $C>0$ is an absolute constant.

As we have indicated above, parallel and concurrent lines are in
correspondence with coset families in $\emph{Aff}(\mathbb{R})$ and hence
Theorem 1.2 is a result on affine transformations. Theorem 1.2 and underlying
ideas were further developed by several authors, having inspired Murphy’s work
[9], which deals with $\emph{Aff}(\mathbb{F})$, $\mathbb{F}$ being
$\mathbb{C}$ or finite. Murphy’s paper, as well as [15], used and combined
ideas of Elekes with Helfgott’s theory of growth in groups, exposed in [6].

Theorem 1.2 is qualitative in the sense that its proofs so far failed to yield
a reasonable quantitative value of $C$, for the proofs involved a graph-
theoretical Balog-Szemerédi-Gowers type argument, which is rather costly for
quantitative estimates.

However, it would not be difficult to derive Theorem 1.2 as a corollary of
Theorem 1.1, provided that there is a nontrivial bound on the energy $E(A)$
therein. Such estimates were obtained in [15] in the special case of
$A=C\times D$ being a Cartesian product. The bound obtained in [15] was that
for some $\delta>0$,

$E(C\times D)\ll|C|^{5/2-\delta}|D|^{3},$ (5)

provided that the sizes of $C$ and $D$ are not vastly different. The
quantitative lower bound on $\delta$ would be, if attempted, quite small. On
the other hand, the above estimate with $\delta=0$ can be regarded in a sense
as a threshold one, and having any $\delta>0$ makes it strong enough to reach
the point at which new information in the arithmetic setting of sum-product
theory is obtained, where incidence configurations involving such Cartesian
products frequently arise. Several such applications were explored in [12],
and also in [15].

##  2 Main results

In this paper we prove the generalisation of the threshold version of estimate
(5) for any finite set $A\subset\emph{Aff}(\mathbb{F})$. This enables us to
develop a number of geometric combinatorics estimates, which so far have been
inaccessible. Our main result is the following:

######  Theorem 2.1.

Let $A$ be a finite set of transformations in the affine group
$\emph{Aff}(\mathbb{F})$ such that no line contains more than $M$ points of
$A$, and no vertical line contains more than $m$ points of $A$. Also, suppose
in positive characteristic $p,$ one has $m|A|\leq p^{2}$. Then,

$\max\\{E(A),E^{*}(A)\\}\ll m^{\frac{1}{2}}|A|^{\frac{5}{2}}+M|A|^{2}.$

Note that if $A$ is taken to be a Cartesian product, this recovers (5), with
$\delta=0$.

We remark that one may consider the asymmetric case of $E(A,B)$ for
$A,B\subset\emph{Aff}(\mathbb{F})$, defined similar to $E(A)$ in (1), only now
$g,u\in A;\,h,v\in B$. Give the notations $m,M$ of Theorem 2.1 subscripts
${}_{A\,\text{or}\,B}$, depending on whether they pertain to $A$ or $B$. The
estimate of Theorem 2.1 then changes to

$E(A,B)\ll
m^{\frac{1}{2}}_{A}|A||B|^{\frac{3}{2}}+\min\\{M_{A},M_{B}\\}|A||B|\,,$

provided that $m_{A}|B|\leq p^{2}$ in positive characteristic. Moreover, if
$\mathbb{F}=\mathbb{F}_{p}$ one can dispense with the $p$-constraint, by using
the asymptotic version of Theorem 3.3, estimate (9). This would add to the
right-hand side of the latter estimate a (crudely estimated) term
$\frac{m_{A}|A||B|^{2}}{p}.$

What is the correct bound for $E(A)$? The example $A=C\times D$, where $C$ is
a geometric progression and $D$ an arithmetic one, with $|D|>|C|$ shows, in
view of the forthcoming equations (7), that in this case $E(A)\gg M|A|^{2}$.
Note that owing to energy quadruples $(g,h,u,v)$ lying in the same (left)
coset of some torus, the upper bound on $E(A)$ should exceed the number of
collinear point triples in $A$. We carefully conjecture that $M|A|^{2}$ may be
the correct asymptotics, possibly with additional subpolynomial factors in
$|A|$, although we do not have evidence for their presence. Something similar
has been conjectured for the concept of bisector energy in [8]. There are
other parallels between bisector energy and our current considerations; see
the appendix for more discussion.

However, the key Lemma 3.2 used in the proof of Theorem 2.1 is itself sharp,
and so is estimate (10) that combined with the latter lemma finishes the proof
of Theorem 2.1. See the forthcoming remark after the proof of Theorem 2.1.

The proof of Theorem 2.1 is geometric, relying on incidence theory. After an
application of Cauchy-Schwarz in the form of (2), it enables an estimate on
growth in $\emph{Aff}(\mathbb{F}_{p})$ which is much stronger than its
algebra-based predecessors in [6, Proposition 4.8], [9, Theorem 27] and even
the geometric [15, Theorem 5] which took advantage of the (sharp) estimate on
the number of directions, determined by a point set in $\mathbb{F}_{p}^{2}$ by
Szőnyi [19].222The conditions of Theorem 2.1 and Corollary 2.2 are identical.
They may appear different as one statement uses geometric and another uses
algebraic language.

######  Corollary 2.2.

Let $A\subset\emph{Aff}(\mathbb{F}_{p})$ have no more than $M$ elements in a
coset of a torus and no more than $m$ elements in a coset of $U$. Suppose
$m|A|\leq p^{2}$. Then

$\min\\{|AA|,|A^{-1}A|\\}\gg m^{-1/2}|A|^{3/2}+M^{-1}|A|^{2}\,.$

In particular, if $|AA|=K|A|,$ then $A$ has $\gg|A|/K^{2}$ elements in a coset
of the unipotent subgroup or $\gg|A|/K$ elements in a coset of a torus.

Theorem 2.1 enables us to establish the following quantitative version of
Theorem 1.2 over general fields.

######  Theorem 2.3.

Let $n,k$ be positive integers, $A\subset\mathbb{F}$ with $|A|=n$ and suppose
there are $k$ non-horizontal and non-vertical lines intersecting $A\times A$
in at least $\alpha n$ points for some $n^{-\frac{1}{2}}\ll\alpha\leq 1$. In
positive characteristic $p$ also assume

$p\gg\max\\{k,\alpha^{-2}n\\}\,.$

Then for $C=12$ if $\mathbb{F}=\mathbb{R}$ and $C=16$ if
$\mathbb{F}=\mathbb{F}_{p}$, we have either
$\bullet~{}$ at least $\gg\alpha^{C}n^{-2}k^{3}$ of these lines are parallel,
and moreover $E^{+}(A)\gg\alpha^{2+C}k^{3}$; or
$\bullet~{}$ at least $\gg\alpha^{C/2}n^{-1}k^{2}$ of these lines are incident
to a common point, and moreover there is $s\in\mathbb{F}$ such that
$E^{\times}(A-s)\gg\alpha^{2+\frac{C}{2}}nk^{2}$.

### Two applications in line geometry

#### Shadow of a point set on two lines

Theorem 2.3 gives an affirmative answer (with a quantitative estimate) to a
conjecture stated in [12, Conjecture 8] – a question that had earlier been
raised by Yufei Zhao. As a matter of fact, it follows from the proof below
that the positive qualitative answer was implicit by the Elekes’ Theorem 1.2.
We give a statement over the reals, although it applies to a general
$\mathbb{F}$, given that $P$ defines $\gg|P|^{2}$ lines.

Given a planar point set $P$ and a line $l$ not incident to $P$, we call the
shadow of $P$ on $l$ the set of points where $l$ meets the lines from $L(P)$ –
the set of lines, generated by pairs of points in $P$. Abusing notation
slightly, we write $L(P)\cap l$ for the shadow of $P$ on $l$.

######  Corollary 2.4.

Let $P\subset\mathbb{R}\mathbb{P}^{2}$ be a finite, sufficiently large set of
points, and let $l_{1}$ and $l_{2}$ be two distinct lines such that
$\ll|P|^{1-\delta_{1}}$ points of $P$ lie on any line passing through the
intersection of $l_{1}$ and $l_{2}$, and $\ll|P|^{1-\delta_{2}}$ points of $P$
lie on any other line, for some $\delta_{1},\delta_{2}>0$. Then

$\max\\{|L(P)\cap l_{1}|,|L(P)\cap
l_{2}|\\}\gg\min\\{|P|^{1+\frac{\delta_{1}}{14}},|P|^{1+\frac{\delta_{2}}{7}}\\}.$

Roughly, a sufficiently non-collinear point set $P$ can have a small shadow on
at most one line.

This question is partially motivated by the sum-product problem. Indeed, if we
take the point set to be $P=\\{(a,a^{2}):a\in A\subset\mathbb{R}\\}$, $l_{1}$
to be the line at infinity and $l_{2}$ to be the $y$-axis, Corollary 2.4, used
as a black box implies that

$|A+A|+|AA|\gg|A|^{1+c}\,,$

with $c=\frac{1}{14}$. This implication is true for sufficiently small sets in
positive characteristic as well, for the role of $\mathbb{R}$ in the proof of
Corollary 2.4 is merely to guarantee that $|L(P)|\gg|P|^{2}$. But in fact, if
$P$ is a parabola as considered above, it’s easy to show that (as a set of
affine transformations) $E(P)\ll|P|^{2}$, and following the argument in the
proof of Corollary 2.4, one gets $c=\frac{1}{7}$ in characteristic zero and
$c=\frac{1}{9}$ in positive characteristic.

A special case of Corollary 2.4 was proven in [12] and used to prove that

$\left|\left\\{\frac{ab-cd}{a-c}:a,b,c,d\in
A\right\\}\right|\gg|A|^{2+\frac{1}{14}},$

for any finite $A\subset\mathbb{R}$. By making suitable choices for the point
set and the two fixed lines, one can also recover quantitatively weaker
versions of several well-known sum-product results, such as lower bounds on
$\max\\{|A+A|,|A^{-1}+A^{-1}|\\}$ and $\max\\{|AA|,|(A+1)(A+1)|\\}$.

We use Corollary 2.4 to point out another curious fact, concerning the shadow
of the Cartesian product $A\times A$ on a single line.

######  Corollary 2.5.

Let $A\subset\mathbb{R}$ be a finite set, and $P=A\times A$. Let $l$ be any
affine line not of the form $y=x$ or $y=-x+k$ for some $k$. Then we have

$|L(A\times A)\cap l|\gg|A|^{2+\frac{1}{14}}.$

#### Quadrangles, rooted on two lines

The other application is as follows. Consider a planar point set $P$, not
incident to the $y$-axis. What is the maximum number of non-trivial
quadrangles $(g,h,u,v)$ with vertices in $P$, so that the two lines connecting
$g,h$ and $u,v$ are parallel, i.e., meet the line at infinity at the same
point, and the two lines, connecting $g,u$ and $h,v$ have the same
$y$-intercept? By non-trivial, we mean here that the four points $g,h,u,v$ do
not lie on the same line. The question and the forthcoming answer applies to
any two lines $l_{1}$, $l_{2}$, not incident to the point set $P$. Let
$\mathcal{Q}(P)$ be the set of quadruples in question.

$g$$u$$h$$v$ Figure 1: An example of a non-degenerate quadrangle
$(g,h,u,v)\in\mathcal{Q}(P)$

Because $|Q(P)|\leq E(P)$, we have the following corollary of Theorem 2.1. In
fact, the two quantities $|\mathcal{Q}(P)|$ and $E(P)$ are equal, modulo
collinear energy quadruples.

######  Corollary 2.6.

Let $P\subset\mathbb{F}^{2}$ such that no line contains more than $M$ points
of $P$, and no vertical line contains more than $m$ points of $P$. Also,
suppose in positive characteristic $p$ one has $m|P|\leq p^{2}$. Then
$|\mathcal{Q}(P)|\leq m^{\frac{1}{2}}|P|^{\frac{5}{2}}+M|P|^{2}.$

We remark that if $P$ is an arithmetic or geometric progression cross itself,
then it can be seen that $|\mathcal{Q}(P)|\gg|P|^{\frac{5}{2}}.$ In the
special case when $P$ does not have more than $\sqrt{|P|}$ collinear points,
Corollary 2.6 yields $|\mathcal{Q}(P)|\ll|P|^{\frac{5}{2}+\frac{1}{4}}.$ Note
that, over the reals and with the condition that no more than $\sqrt{|P|}$ are
collinear, one has the bound $\ll|P|^{3}\log|P|$ for the number of quadruples
$(g,h,u,v)$ such that the direction from $g$ to $h$ is equal to that from $u$
to $v$. See [14, Theorem 5]. After a projective transformation, the same bound
holds for the number of quadruples $(g,h,u,v)$ such that the line connecting
$g$ and $u$ has the same $y$-intercept as the line connecting $h$ and $v$. In
view of the above lower bound, the events of the lines along pairs of opposite
sides of the quadrangles in $\mathcal{Q}(P)$ meeting the lines $l_{1}$ and
$l_{2}$ at the same point on the latter lines cannot be regarded as
independent.

##  3 Proofs

###  3.1 Proof of Theorem 2.1

We adopt the convention that for any $x\in\emph{Aff}(\mathbb{F})$,
$x=(x_{1},x_{2})$.

###### Proof.

We prove the estimate for the quantity $E(A)$, the proof for $E^{*}(A)$
follows from the inequality $E^{*}(A)\leq E(A)$ from [17]. Alternatively the
forthcoming proof has an isomorphic version with $E^{*}(A)$ in place of
$E(A)$.

Set

$Q(A):=\\{(g,h,u,v)\in A^{4}:g^{-1}\circ h=u^{-1}\circ v\\},$ (6)

so that $E(A)=|Q(A)|$. The first lemma gives a convenient decomposition of
this set.

######  Lemma 3.1.

Let $A\subset G$ be finite. Then we have

$E(A)=\sum_{C}|\\{(g,h,u,v)\in Q(A):g_{1}v_{1}=C=h_{1}u_{1}\\}|.$

###### Proof.

Suppose that $g^{-1}\circ h=u^{-1}\circ v$. Then

$(g_{1}^{-1},g_{2}g_{1}^{-1})\circ(h_{1},h_{2})=(u_{1}^{-1},u_{2}u_{1}^{-1})\circ(v_{1},v_{2}).$

Calculating the first coordinates then gives

$h_{1}g_{1}^{-1}=v_{1}u_{1}^{-1}$

which rearranges to $g_{1}v_{1}=h_{1}u_{1}$. ∎

Let $Q_{C}$ denote the set of solutions corresponding to the value $C$ in the
decomposition above, that is,

$Q_{C}:=|\\{(g,h,u,v)\in Q(A):g_{1}v_{1}=C=h_{1}u_{1}\\}|.$

Also define the set

$\mathcal{C}_{C}:=\\{(g,v)\in A\times A:g_{1}v_{1}=C\\}.$

We bound $Q_{C}$ by the following lemma.

######  Lemma 3.2.

For any $C\in\mathbb{F}^{*}$, under the constraint that $m|A|\leq p^{2}$ in
positive characteristic, one has

$Q_{C}\ll|\mathcal{C}_{C}|^{3/2}+M|\mathcal{C}_{C}|.$

###### Proof of Lemma 3.2.

By considering the line equation $g^{-1}\circ h=u^{-1}\circ v$ and expanding
out, we have the equations

$g_{1}v_{1}=h_{1}u_{1}=C,\qquad\frac{h_{2}-g_{2}}{g_{1}}=\frac{v_{2}-u_{2}}{u_{1}}.$
(7)

The second equation can be written as

$u_{2}g_{1}-u_{1}g_{2}-g_{1}v_{2}+u_{1}h_{2}=0\,.$ (8)

We therefore map $\mathcal{C}_{C}\times\mathcal{C}_{C}$ to
$\mathbb{P}^{3}\times\mathbb{P}^{3*}$ (where $\mathbb{P}^{3}$ is the three-
dimensional projective space over $\mathbb{F}$ and $\mathbb{P}^{3*}$ is its
dual) as

$(g,v)\to(g_{1}:g_{2}:g_{1}v_{2}:1),\qquad(u,h)\to(u_{2}:-u_{1}:-1:u_{1}h_{2})\,.$

The map is injective. Indeed, suppose we have
$(g_{1},g_{2},g_{1}v_{2})=(g_{1}^{\prime},g_{2}^{\prime},g_{1}^{\prime}v_{2}^{\prime})$.
Then we must have $(g_{1},g_{2})=(g_{1}^{\prime},g_{2}^{\prime})$, and since
$g_{1}v_{1}=g_{1}^{\prime}v_{1}^{\prime}=C$, we must also have
$v_{1}=v_{1}^{\prime}$. The last coordinate then gives
$g_{1}v_{2}=g_{1}^{\prime}v_{2}^{\prime}\implies v_{2}=v_{2}^{\prime}$.

Thus equation (8) can be viewed as a point plane incidence between the set of
points

$P_{C}=\\{(g_{1},g_{2},g_{1}v_{2}):(g,v)\in\mathcal{C}_{C}\\}$

and the set of planes

$\Pi_{C}=\\{\pi_{u,h}:(u,h)\in\mathcal{C}_{C}\\},$

where $\pi_{u,h}$ is the plane given by $u_{2}x-u_{1}y-z+u_{1}h_{2}=0.$ Note
that we have $|P_{C}|=|\Pi_{C}|=|\mathcal{C}_{C}|$.

This calls for using the point-plane incidence bound from [13]:

######  Theorem 3.3 (Rudnev).

Let $\mathbb{F}$ be a field, and let $P$ and $\Pi$ be finite sets of points
and planes respectively in $\mathbb{P}^{3}$. Suppose that $|P|\leq|\Pi|$, and
that $|P|\ll p^{2}$ if the characteristic $p\neq 2$ of $\mathbb{F}$ is
positive. Let $k$ be the maximum number of collinear points in $P$. Then the
number of incidences satisfies

$I(P,\Pi)\ll|\Pi||P|^{1/2}+k|\Pi|$

Note that if $\mathbb{F}=\mathbb{F}_{p}$, one can dispense with the
$p$-constraint in Theorem 3.3, replacing its estimate by the following
asymptotic version (see, e.g. [16, Lemma 6]):

$I(P,\Pi)-\frac{|\Pi||P|}{p}\ll|\Pi||P|^{1/2}+k|\Pi|\,.$ (9)

We claim that we can set $k=M$ in our application of Theorem 3.3. Indeed,
suppose that a line contains more than $M$ points from $P_{C}$, each of the
form $(g_{1},g_{2},g_{1}v_{2})$. If this line is not of the form
$\\{(x_{0},y_{0},z):z\in\mathbb{F}\\}$ then project using the map
$\pi(x,y,z)=(x,y,0)$ to get a line in the plane which contains more than $M$
points of the form $(g_{1},g_{2})$. This contradicts the assumption that $A$
contains no more than $M$ points on a line.

If the line is of the form $\\{(x_{0},y_{0},z):z\in\mathbb{F}\\}$ and it
contains more than $M$ points from $P_{C}$, it follows that there are more
than $M$ points $(v_{1},v_{2})$ in $A$, all satisfying $v_{1}=C/x_{0}$.

To conclude the proof we apply Theorem 3.3 to the point-plane arrangement
$P_{C}$ and $\Pi_{C}$. We note the trivial (yet easily seen to be achievable)
supremum bound

$|\mathcal{C}_{C}|\leq m|A|\,,$ (10)

which determines the constraint in positive characteristic, and then have

$|Q_{C}|\leq I(P_{C},\Pi_{C})\ll|\mathcal{C}_{C}|^{3/2}+M|\mathcal{C}_{C}|\,,$

as required. ∎

The claim of Theorem 2.1 now follows immediately. Apply Lemma 3.1 and bound
each summand by Lemma 3.2 to get

$E(A)\leq\sum_{C}\left(|\mathcal{C}_{C}|^{3/2}+M|\mathcal{C}_{C}|\right).$

Applying the $L^{\infty}$ bound (10) and the $L^{1}$ identity

$\sum_{C}|\mathcal{C}_{C}|=|A|^{2}\,$

completes the proof. ∎

Lemma 3.2 cannot be improved without additional restrictions. Indeed, consider
the case $A=[n]\times[n]$. If we take $C=q$ for some prime $q\leq n$, we have
$|\mathcal{C}_{C}|=2n^{2}$. We also have $M=n$ (hence $M$ is a constant
multiple of $|\mathcal{C}_{C}|^{1/2}$ \- the regime in which the two terms in
Lemma 3.2 are of the same order of magnitude). Lemma 3.2 tells us that
$Q_{C}\ll n^{3}$. On the other hand, every solution to the additive energy
equation

$b_{1}+b_{2}=b_{3}+b_{4},\,\,\,\,b_{i}\in\\{1,\dots,n\\}$

gives a contribution to $Q_{C}$, and so $Q_{C}\gg n^{3}$.

In fact, if we instead take $C$ to be any element of the product set of $[n]$,
then since $C$ has $\ll n^{\epsilon}$ representations as a product we have
$|\mathcal{C}_{C}|\ll n^{2+\epsilon}$, and so the bound is near-optimal for
all $C$ for this choice of $A$.

Moreover, estimate (10) alone is also sharp if one simply takes $A=C\times D$,
where $C$ is a geometric progression.

###  3.2 Proof of Theorem 2.3

###### Proof.

We apply Theorems 1.1 and 2.1. The constraints on $\alpha$ in the statement of
Theorem 2.3 have been chosen to ensure dominance of the first terms in the
incidence bounds (3), (4). Let $L$ be the set of $k$ lines described in the
statement of Theorems 2.1 and 2.3.

For $\mathbb{F}=\mathbb{R}$ an application of Theorems 1.1 and 2.1 combined
with the fact that $\alpha n\geq C^{\prime}n^{1/2}$ gives

$\alpha kn\leq I(A\times A,L)\ll
n^{7/6}\left(m^{1/12}k^{3/4}+M^{1/6}k^{2/3}\right)\,,$

so either $m\gg\alpha^{12}n^{-2}k^{3}$ or $M\gg\alpha^{6}n^{-1}k^{2}$. It
follows that either $\gg\alpha^{12}n^{-2}k^{3}$ lines, as elements of $G$, lie
in the same coset of $U$, so they are parallel, or $\gg\alpha^{6}n^{-1}k^{2}$
lines lie in the same coset of some torus, so they are concurrent.

Similarly, for $\mathbb{F}=\mathbb{F}_{p}$, Theorems 1.1 and 2.1 combined (the
condition $k\leq p$ ensures that $mk\leq p^{2}$) with the fact that $\alpha
n\geq C^{\prime}n^{1/2}$ and $p\geq\alpha^{-2}n$ give

$\alpha kn\ll n^{9/8}\left(m^{1/16}k^{13/16}+M^{1/8}k^{3/4}\right)\,,$

hence either $\gg\alpha^{16}n^{-2}k^{3}$ lines are parallel or
$\gg\alpha^{8}n^{-1}k^{2}$ lines are concurrent.

To address the second claim of the first bullet point, parameterise the set of
parallel lines by by $(\gamma,\beta)$, where $\gamma\neq 0$ is the fixed
slope, and denote by $B\subseteq\mathbb{F}_{p}$ the set of all such $\beta$.
By the assumption that we are in the case of the first bullet point, we have

$|B|\gg\alpha^{C}n^{-2}k^{3}\,,$

where $C=12$ or $C=16.$ Hence, applying Cauchy-Schwarz twice, one has (with
$r_{F(A)}(\beta)$ standing for the number of realisations of $\beta$ as a
value of a function $F$, with several variables in $A$, and $A(x)$ being the
indicator function of $A$):

$\displaystyle\alpha n|B|$ $\displaystyle\leq\sum_{\beta\in B}\sum_{x\in
A}A(\gamma x+\beta)=\sum_{\beta\in B}r_{A-\gamma
A}(\beta)\leq\sqrt{|B|}\left(\sum_{\beta\in B}r^{2}_{A-\gamma
A}(\beta)\right)^{1/2}$ (11) $\displaystyle\leq\sqrt{|B|}\sqrt{E^{+}(A,\gamma
A)}\leq\sqrt{|B|}\sqrt{E^{+}(A)}\,,$

proving the second claim of the first bullet point.

Similarly in the case of $\gg\alpha^{C/2}n^{-1}k^{2}$ lines being concurrent
at some point $(x_{0},y_{0})$, we parameterise them by their slopes, once
again denoted as $\beta=\frac{y-y_{0}}{x-x_{0}}$ in a set $B$, with
$|B|\gg\alpha^{C/2}n^{-1}k^{2}$.

As in (11) above, we have

$\alpha n|B|\leq\sum_{\beta\in
B}r_{\frac{A-y_{0}}{A-x_{0}}}(\beta)\leq\sqrt{|B|}({E^{\times}}(A-x_{0}){E^{\times}}(A-y_{0}))^{\frac{1}{4}}\,,$

proving the second claim of the second bullet point.

∎

###  3.3 Proof of Corollaries 2.4, 2.5, 2.6

In this section we first apply Theorem 2.1 to prove Corollary 2.4, whence we
deduce Corollary 2.5. At the end we prove Corollary 2.6.

###### Proof of Corollary 2.4.

Let $l_{1}$, $l_{2}$ be distinct arbitrary lines, and let $P$ be a set of
points in the plane with no more than $|P|^{1-\delta_{1}}$ points of $P$ on a
line through the intersection of $l_{1}$ and $l_{2}$, and no more than
$|P|^{1-\delta_{2}}$ points on any other line. By assumption, no more than
$|P|^{1-\delta_{1}}$ points lie on $l_{1}$ or $l_{2}$, so we may remove such
points and be left with a positive proportion of $P$. We apply a projective
transformation $\pi$ to send $l_{1}$ to the $y$-axis $l_{y}$, and $l_{2}$ to
the line at infinity $l_{\infty}$. Defining $P^{\prime}:=\pi(P)$, we have

$|L(P)\cap l_{1}|=|L(P^{\prime})\cap l_{y}|,\quad|L(P)\cap
l_{2}|=|L(P^{\prime})\cap l_{\infty}|$

which follows since projective transformations preserve incidence structure.
Let $T$ denote the set $L(P^{\prime})\cap l_{y}$ and $S$ denote the set
$L(P^{\prime})\cap l_{\infty}$. Note that lines through the intersection of
$l_{1}$ and $l_{2}$ have been mapped to vertical lines. We claim that we have

$I(P^{\prime},L(P^{\prime}))\leq I(S\times T,P^{\prime})$

where on the right side $P^{\prime}$ is being considered as a subset of the
affine plane $\emph{Aff}(\mathbb{R})$ corresponding to a set of lines. The
claim follows by taking a point $(p_{1},p_{2})\in P^{\prime}$ lying on some
line $y=sx+t$ coming from $L(P^{\prime})$. By definition we must have
$(s,t)\in S\times T$, and therefore this incidence may be viewed as the point
$(s,t)\in S\times T$ lying on the line $y=-p_{1}x+p_{2}$, proving the claim.

We bound these incidences using Theorem 1.1, applied to the point set $S\times
T$ and the line set $P^{\prime}$ (strictly we are applying it to the line set
given by $(-p_{1},p_{2})$ such that $(p_{1},p_{2})\in P^{\prime}$, but this
changes nothing with respect to the collinearity conditions). The lines in
$P^{\prime}$ are non-vertical and non-horizontal since we removed any points
lying on $l_{1}$ or $l_{2}$ in $P$, which implies that $P^{\prime}$ has only
affine points lying off the $y$ axis. Furthermore, $P^{\prime}$ has no more
than $|P|^{1-\delta_{1}}$ points on any vertical line, and no more than
$|P|^{1-\delta_{2}}$ points on any other line. We may bound the energy
$E(P^{\prime})$ using Theorem 2.1. Plugging in
$m\ll|P^{\prime}|^{1-\delta_{1}}$, $M\ll|P^{\prime}|^{1-\delta_{2}}$, we have
$E(P^{\prime})\ll|P^{\prime}|^{3-\frac{\delta_{1}}{2}}+|P^{\prime}|^{3-\delta_{2}}$,
implying that

$\displaystyle I(S\times T,P^{\prime})$
$\displaystyle\ll|T|^{1/2}|S|^{2/3}E(P^{\prime})^{1/6}|P^{\prime}|^{1/3}+|T|^{1/2}|P^{\prime}|$
$\displaystyle\ll|T|^{1/2}|S|^{2/3}\left(|P^{\prime}|^{3-\frac{\delta_{1}}{2}}+|P^{\prime}|^{3-\delta_{2}}\right)^{1/6}|P^{\prime}|^{1/3}+|T|^{1/2}|P^{\prime}|$
$\displaystyle\ll|T|^{1/2}|S|^{2/3}|P^{\prime}|^{\frac{5}{6}-\frac{\delta_{1}}{12}}+|T|^{1/2}|S|^{2/3}|P^{\prime}|^{\frac{5}{6}-\frac{\delta_{2}}{6}}+|T|^{1/2}|P^{\prime}|.$

Since each line in $L(P^{\prime})$ intersects $P^{\prime}$ in at least two
places and by Beck’s theorem [1] we have $|L(P^{\prime})|\gg|P^{\prime}|^{2}$,
we may bound below by

$I(S\times T,P^{\prime})\geq
I(P^{\prime},L(P^{\prime}))\geq|L(P^{\prime})|\gg|P^{\prime}|^{2}.$

Putting this all together, we have

$|P^{\prime}|^{2}\ll|T|^{1/2}|S|^{2/3}|P^{\prime}|^{\frac{5}{6}-\frac{\delta_{1}}{12}}+|T|^{1/2}|S|^{2/3}|P^{\prime}|^{\frac{5}{6}-\frac{\delta_{2}}{6}}+|T|^{1/2}|P^{\prime}|.$

If the third term dominates we have a stronger result, so we may assume one of
the first terms dominates, giving

$\max\\{|L(P)\cap l_{1}|,|L(P)\cap
l_{2}|\\}=\max\\{|S|,|T|\\}\gg\min\\{|P^{\prime}|^{1+\frac{\delta_{1}}{14}},|P^{\prime}|^{1+\frac{\delta_{2}}{7}}\\}.$

This gives the result since $|P|=|P^{\prime}|$. ∎

To summarise the proof above, let $P^{\prime}\subseteq P$ be the set of Beck
points of $P$, each incident to $\gg|P|$ lines in $L(P)$. Then $P^{\prime}$
can be viewed as a set of $\gg|P|$-rich affine transformations in the grid
$S\times T$, and the claim of Corollary 2.4 follows from the quantitative
Elekes theorem, Theorem 2.3. This means that the qualitative answer to the
question was, in fact, implied in the original Elekes’ Theorem 1.2.

###### Proof of Corollary 2.5.

Let $A\subseteq\mathbb{R}$ be a finite set, and $l$ an affine line not of the
form $y=-x+k$ for some $k$, or the line $y=x$. We shall prove that $|L(A\times
A)\cap l|\gg|A|^{2+\frac{1}{14}}$.

Let $\gamma$ denote the transformation given by reflection in the line $y=x$.
The Cartesian product $A\times A$ is sent to itself under this reflection.
Furthermore, by the restriction placed on the line $l$, we must have that
$l^{\prime}:=\gamma(l)\neq l$.

Since $A\times A$ is symmetric under $\gamma$, it follows that $L(A\times A)$
is also symmetric under $\gamma$, and moreover that the number of
intersections of $L(A\times A)$ with $l$ is precisely the number of
intersections of $L(A\times A)$ with $l^{\prime}$, i.e. $|L(A\times A)\cap
l|=|L(A\times A)\cap l^{\prime}|$. An application of Corollary 2.4 with
$\delta_{1}=\delta_{2}=1/2$ then gives

$|L(A\times A)\cap l|=\max\\{|L(A\times A)\cap l|,|L(A\times A)\cap
l^{\prime}|\\}\gg|A|^{2+\frac{1}{14}}.$

∎

Note that one may consider a more general version of Corollary 2.5, applying
to a point set in $\mathbb{R}^{2}$ with a symmetry.

###### Proof of Corollary 2.6.

Let us show that if a quadrangle $(g,h,u,v)$ is in $\mathcal{Q}(P)$, then it
also satisfies the energy equation, that is $(g,h,u,v)\in Q(P)$, defined by
(6). Moreover, the sets $Q(P),\mathcal{Q}(P)$ are equal up to the case of
degenerate members $(g,h,u,v)\in Q(P)$, where the points $g,h,u,v$ are
collinear, lying as elements of $\emph{Aff}(\mathbb{F})$ in the same coset of
some torus $T$ (or $U$). In the sequel we will say torus and just use the
notation $T$ for either case, unless there is something special as to $U$.

For the first implication, if

$g^{-1}\circ h=u^{-1}\circ v=t$

then also

$g\circ u^{-1}=h\circ v^{-1}=t^{\prime}\,.$

Then $t,t^{\prime}$, unless they are the identity, which corresponds to a
trivial count, lie respectively in some unique tori $T,T^{\prime}$. If say
$t\in T$, this means $g,h$ lie in some left coset of $T$ and $u,v$ in some
other left coset of $T$. This means the lines connecting the corresponding the
sides $gh$ and $uv$ of the quadrangle $g,h,v,u$ are parallel. Similarly $g,u$
and $h,v$ lie in two distinct right cosets of $T^{\prime}$, and hence the
sides $gu$ and $hv$ of the quadrangle $g,h,v,u$ have the same $y$-intercept.

The converse is slightly less obvious. Suppose, $(g,h,u,v)\in\mathcal{Q}(P)$.
We therefore have two tori $T,T^{\prime}$ defined respectively by the equal
slope of the lines $gh$ and $uv$, and the equal $y$-intercept of the lines
$gu$ and $hv$. Hence, there are two left cosets of $T$, with representatives
$c_{1},c_{2}$, and elements $n_{1},n_{2},\nu_{1},\nu_{2}\in T$, as well as two
right cosets of $T^{\prime}$, with representatives $s_{1},s_{2}$, and elements
$m_{1},m_{2},\mu_{1},\mu_{2}\in T^{\prime}$, so that

$g=c_{1}n_{1}=m_{1}s_{1},\qquad h=c_{1}n_{2}=\mu_{1}s_{2},\qquad
u=c_{2}\nu_{1}=m_{2}s_{1},\qquad v=c_{2}\nu_{2}=\mu_{2}s_{2}\,.$

This implies, using the Abelian notation within the commutative
$T,T^{\prime}$, that in particular

$(v^{-1}\circ u)+(g^{-1}\circ
h)=n_{2}-n_{1}+\nu_{1}-\nu_{2}=s^{-1}\circ(m_{2}-m_{1}+\mu_{1}-\mu_{2})\circ
s^{-1}\,,$

where $s=s_{1}$ or $s=s_{2}$. It follows that
$m_{2}-m_{1}+\mu_{1}-\mu_{2}=n_{2}-n_{1}+\nu_{1}-\nu_{2}$ equals identity.
Indeed, suppose not. If say the left-hand side of the latter equation is in
the normal subgroup $U$, then so is the right-hand side. This means that the
quadrangle $(g,h,u,v)$ has four vertical sides, namely the points $g,h,u,v$
lie on the same vertical line. The rest of the tori are each other’s
conjugates, meeting at the identity only. Then one has $s_{1}=s_{2}$, once
again making the points $g,h,u,v$ collinear.

∎

##  4 Appendix: Bisector Energy, Beck’s Theorem and Point-Plane Incidences

The approach taken to bounding affine energy in this paper was inspired by a
series of works bounding the bisector energy of a point set. Given a finite
point set $P\subset\mathbb{F}$, its bisector energy is defined to be the
number of solutions to

$B(p,q)=B(s,t),\,\,\,\,\,\,\,\,p,q,s,t\in P,$ (12)

where $B(p,q)$ denotes the perpendicular bisector of $p$ and $q$.

Let $\mathcal{B}(P)$ denote the bisector energy of a set $P$. The situation is
rather similar to what we face when trying to bound the affine group energy. A
trivial bound is $\mathcal{B}(P)\leq|P|^{3}$, and taking the point set to be
equidistributed on a line or circle realises $\mathcal{B}(P)\gg|P|^{3}$. So,
as with our situation, in order to make progress it is necessary to impose
some conditions on the original set, ruling out highly collinear or co-
circular situations.

Let $M$ denote the maximum number of points from $P$ on a single line or
circle. The following bound was proven for $P\subset\mathbb{R}^{2}$ in [7]:

$\mathcal{B}(P)\ll|P|^{5/2}+M|P|.$ (13)

The proof of (13) used Beck’s Theorem [1], and since this is not known to hold
in the finite field setting, (13) does not immediately translate to
$\mathbb{F}_{p}^{2}$. However, in [11] (see also [10] that superseded it) , it
was shown that Beck’s Theorem was not necessary, and a more direct approach
was given, making use of Theorem 3.3. A similar situation occurred in drafting
this paper: an earlier draft used an application of Beck’s Theorem to prove
Lemma 3.2, based on the approach of [7], before we discovered that the proof
can be simplified and generalised via a fairly straightforward application of
Theorem 3.3.

These two instances have highlighted a connection between Beck’s Theorem and
Theorem 3.3. To illustrate this connection, we conclude the paper with a new,
short proof of Theorem 3.3 in the Euclidean setting which uses only Beck’s
Theorem as an input. This is a considerable simplification on the original
proof from [13], which used deeper techniques and built on the incidence
theory developed by Guth and Katz [5].

###### Proof of Theorem 3.3 over $\mathbb{R}^{3}$.

Recall that the aim is to prove that

$I(P,\Pi)\ll|\Pi||P|^{1/2}+|P||\Pi|^{1/2}+k|\Pi|$ (14)

holds for any set $P$ and $\Pi$ of points and planes in $\mathbb{R}^{3}$
respectively, such that no more than $k$ points from $P$ are contained in a
single line. By Cauchy-Schwarz,

$\displaystyle I(P,\Pi)$ $\displaystyle=\sum_{\pi\in\Pi}|\\{p\in
P:p\in\pi\\}|\leq|\Pi|^{1/2}\left(\sum_{\pi\in\Pi}|\\{p\in
P:p\in\pi\\}|^{2}\right)^{1/2}$
$\displaystyle=|\Pi|^{1/2}\left(I(P,\Pi)+\sum_{\pi\in\Pi}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|\right)^{1/2}.$

If the first term in the bracket dominates then we have $I(P,\Pi)\ll|\Pi|$,
which is much stronger than required. Therefore we can disregard it, and
assume that

$I(P,\Pi)\ll|\Pi|^{1/2}\left(\sum_{\pi\in\Pi}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|\right)^{1/2}.$ (15)

In each plane $\pi$, Beck’s Theorem tells us that there are two possibilities

  * (i)

$\gg|\pi\cap P|^{2}$ of the pairs of distinct points from $\pi\cap P$ are
contained on a single line,

  * (ii)

$\gg|\pi\cap P|^{2}$ of the pairs of distinct points from $\pi\cap P$ are
contained on lines which support less than $C$ points from $\pi\cap P$, for
some absolute constant $C$.

Let $\Pi_{1}$ be the set of all planes of type (i), and $\Pi_{2}$ those of
type (ii). We seek to bound

$\displaystyle\sum_{\pi\in\Pi}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|$
$\displaystyle\leq\sum_{\pi\in\Pi_{1}}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|$
$\displaystyle+\sum_{\pi\in\Pi_{2}}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|.$ (16)

Since there are no more than $k$ points from $P$ on any line, the first term
above contributes at most $k^{2}|\Pi|$. For the second term, we use Cauchy-
Schwarz to obtain

$\displaystyle\sum_{\pi\in\Pi_{2}}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|$ $\displaystyle\leq C^{2}\sum_{l\in L(P)}|\\{\pi\in\Pi:l\in\pi\\}|$
$\displaystyle\ll|L(P)|^{1/2}\left(\sum_{l\in
L(P)}\sum_{\pi_{1},\pi_{2}\in\Pi:l\in\pi_{1}\cap\pi_{2}}1\right)^{1/2}$
$\displaystyle\ll|P|\left(|\Pi|^{2}+\sum_{l\in
L(P)}|\\{\pi\in\Pi:l\in\pi\\}|\right)^{1/2}$

If the second term is dominant then we get

$\sum_{\pi\in\Pi_{2}}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|\ll|P|^{2}.$

Otherwise the first term is dominant and we get $|P||\Pi|$ on the right hand
side. Therefore,

$\sum_{\pi\in\Pi_{2}}|\\{(p_{1},p_{2})\in(P\cap\pi)^{2}:p_{1}\neq
p_{2}\\}|\ll|P|(|P|+|\Pi|).$

Combining this with (15) and (4), we finally conclude that

$I(P,\Pi)\ll k|\Pi|+|P||\Pi|^{1/2}+|P|^{1/2}|\Pi|,$

as required. ∎

One may compare the proof of (14) above with the work of Elekes and Toth [3],
who also proved a point-plane incidence bound in $\mathbb{R}^{3}$ with a
slightly different notion of non-degeneracy. The bound in [3] is better in the
asymmetric case. Rather than using Beck’s Theorem [1], the proof in [3] uses
the full strength of the Szemerédi-Trotter Theorem [18].

Acknowledgements. The project begun while the second and third listed authors
visited The University of Georgia. The first listed author is supported by the
NSF Award 1723016 and gratefully acknowledges the support from the RTG in
Algebraic Geometry, Algebra, and Number Theory at the University of Georgia,
and from the NSF RTG grant DMS-1344994. The second and fourth listed authors
were partially supported by the Austrian Science Fund FWF Project P 30405-N32.
The third listed author was partially supported by the Leverhulme grant
RPG-2017-371. We thank Brendan Murphy and Adam Sheffer for helpful discussions
and also Ilya Shkredov for his feedback on an earlier draft of the paper.
Finally, we thank an anonymous referee for their comments and corrections.

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Giorgis Petridis
Department of Mathematics, University of Georgia, Athens, GA 30602, USA.
<EMAIL_ADDRESS>
Oliver Roche-Newton
Johann Radon Institute for Computational and Applied Mathematics
69 Altenberger Straße, Linz 4040, Austria.
<EMAIL_ADDRESS>
Misha Rudnev
School of Mathematics, Fry Building, Bristol BS8 1TH, UK.
<EMAIL_ADDRESS>
Audie Warren
Johann Radon Institute for Computational and Applied Mathematics
69 Altenberger Straße, Linz 4040, Austria.
<EMAIL_ADDRESS>