Some remarks on products of sets in the Heisenberg group and in the affine group

TL;DR

This paper explores the behavior of product sets in the Heisenberg and affine groups over prime fields, providing new growth results and applications to Freiman's isomorphism in nonabelian groups.

Contribution

It introduces novel results on product set growth in nonabelian groups and applies these findings to Freiman's isomorphism problem.

Findings

New growth results for product sets in the Heisenberg group

New growth results for product sets in the affine group over prime fields

Application to Freiman's isomorphism in nonabelian groups

Abstract

We obtain some new results on products of large and small sets in the Heisenberg group as well as in the affine group over the prime field. Also, we derive an application of these growth results to Freiman's isomorphism in nonabelian groups.

Equations

$$A+B:=\{a+b:a\in A,b\in B\},$$

$$AB:=\{ab:a\in A,b\in B\},$$

$$A/B:=\{a/b:a\in A,b\in B,b\ne 0\},$$

$$\max \{|A+A|,|AA|\}\gg |A{|}^{2-ϵ}.$$

$${\mathcal{H}}\supseteq\mathcal{A}:=\left\{\left({\begin{array}[]{ccc}1&a&0\\ 0&1&b\\ 0&0&1\\ \end{array}}\right)~{}:~{}a,b\in A\right\}\,.$$

$$|\mathcal{AA}|\ge |A{|}^2\max \{|AA|,|A+A|\}{\gg }_{\epsilon }|A{|}^{3+7/22-\epsilon }.$$

$$|\mathcal{AA}|\gg |\mathcal{A}{|}^{7/4+c},$$

$$|\mathcal{AA}|\gg \min \{|\mathcal{A}{|}^{7/4},p|\mathcal{A}|\}.$$

$$\mathcal{A}=\{[x,y,z]:x\in X,y\in Y,z\in Z\}\subseteq {\mathcal{H}}_n$$

$$|\mathcal{A}|>|{\mathcal{H}}_n{|}^{3/4+\epsilon },$$

$$a_1^{{\epsilon }_1}\dots a_s^{{\epsilon }_s}=b_1^{{\epsilon }_1}\dots b_s^{{\epsilon }_s}⟹\rho (a_1{)}^{{\epsilon }_1}\dots \rho (a_s{)}^{{\epsilon }_s}=\rho (b_1{)}^{{\epsilon }_1}\dots \rho (b_s{)}^{{\epsilon }_s}.$$

$${\mathsf{E}}^{+}(A,B)=|\{(a_1,a_2,b_1,b_2)\in A\times A\times B\times B:a_1+b_1=a_2+b_2\}|.$$

$${\mathsf{T}}_k^{+}(A):=|\{(a_1,\dots ,a_k,a_1^{\prime },\dots ,a_k^{\prime })\in A^{2k}:a_1+\cdots +a_k=a_1^{\prime }+\cdots +a_k^{\prime }\}|.$$

$${\mathsf{E}}^{+}(A,B)=\sum _x{r}_{A+B}^2(x)=\sum _x{r}_{A-B}^2(x)=\sum _x{r}_{A-A}(x)r_{B-B}(x).$$

$$\mathsf{E}(A,B)=|\{(a_1,a_2,b_1,b_2)\in A\times A\times B\times B:a_1{b}_1^{-1}=a_2{b}_2^{-1}\}|.$$

$$\hat{f}(\pi )=\sum _{g\in \mathbf{G}}f(g)\pi (g).$$

$$f(g)=\frac{1}{|\mathbf{G}|}\sum _{\pi \in \hat{\mathbf{G}}}{d}_{\pi }⟨\hat{f}(\pi ),\pi (g^{-1}){⟩}_{HS},$$

$$\sum _{g\in \mathbf{G}}|f(g){|}^2=\frac{1}{|\mathbf{G}|}\sum _{\pi \in \hat{\mathbf{G}}}{d}_{\pi }\Vert \hat{f}(\pi ){\Vert }_{HS}^2.$$

$$\widehat{f\ast g}(\pi )=\hat{f}(\pi )\hat{g}(\pi ),$$

$$(f\ast g)(x)=\sum _{y\in \mathbf{G}}f(y)g(y^{-1}x).$$

$${\cal I}(r,l)=\left\{\begin{array}[]{ll}1&\textrm{if }p\in l\\ 0&\textrm{otherwise.}\end{array}\right.$$

$$\mathcal{I}(\mathcal{P},\mathcal{L})\le \min \{|\mathcal{P}{|}^{1/2}|\mathcal{L}|+|\mathcal{P}|,|\mathcal{L}{|}^{1/2}|\mathcal{P}|+|\mathcal{L}|\},$$

$$\left|\sum _{r\in l}\mathcal{I}(r,l)f(r)g(l)\right|\le \sqrt{p}\Vert f{\Vert }_2\Vert g{\Vert }_2,$$

$$\mathcal{I}(\mathcal{P},\mathcal{L})-\frac{|A||B||\mathcal{L}|}{p}\ll |A{|}^{3/4}|B{|}^{1/2}|\mathcal{L}{|}^{3/4}+|\mathcal{L}|+|A||B|.$$

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

$$[x,y,z]=[\vec{x},\vec{y},z]:=\left({\begin{array}[]{ccc}1&\vec{x}&z\\ \vec{0}^{n}&\vec{1}^{n}&\vec{y}\\ \vec{0}^{n}&0&1\\ \end{array}}\right)\,.$$

$$[x,y,z]\cdot [x^{\prime },y^{\prime },z^{\prime }]=[x+x^{\prime },y+y^{\prime },z+z^{\prime }+x{y}^{\prime }],$$

$$[x,y,z{]}^{-1}=[-x,-y,-z+xy].$$

$$\left({\begin{array}[]{c}X\\ Y\\ \end{array}}\right)=\left({\begin{array}[]{cc}1&a\\ 0&1\\ \end{array}}\right)\left({\begin{array}[]{c}x\\ y\\ \end{array}}\right)+\left({\begin{array}[]{c}b\\ c\\ \end{array}}\right)=\left({\begin{array}[]{c}x+ay+b\\ y+c\\ \end{array}}\right)\,,$$

$$f([x,y,z])=\frac{{\delta }_f(x,y)}{|\mathbb{F}|}+\frac{⟨\hat{f}(\pi ),\pi ([x,y,z{]}^{-1}){⟩}_{HS}}{|\mathbb{F}{|}^{n+1}},$$

Full text

Some remarks on products of sets in the Heisenberg group and in the affine group

††This work is supported by the Russian Science Foundation under grant 19–11–00001.

Shkredov I.D

Annotation.

*We obtain some new results on products of large and small sets in the Heisenberg group as well as in the affine group over the prime field. Also, we derive an application of these growth results to Freiman’s isomorphism in nonabelian groups. *

1 Introduction

Let $p$ be an odd prime number, and ${\mathbb{F}}_{p}$ be the finite field. Given two sets $A,B\subset{\mathbb{F}}_{p}$, define the sumset, the product set and the quotient set of $A$ and $B$ as

[TABLE]

[TABLE]

and

[TABLE]

correspondingly. This paper is devoted to the so–called sum–product phenomenon, which says that either the sumset or the product set of a set must be large up to some natural algebraic constrains. One of the strongest form of this principle is the Erdős–Szemerédi conjecture [3], which says that for any sufficiently large set $A$ of real numbers and an arbitrary $\epsilon>0$ one has

[TABLE]

The best up to date results in the direction can be found in [19] and in [14] for ${\mathbb{R}}$ and ${\mathbb{F}}_{p}$, respectively. Basically, in this paper we restrict ourselves to the case of the finite fields only.

It is well–known that the sum–product phenomenon is connected with growth in the group of affine transformations, see, e.g., [11], [16]. Another group which is connected to this area is the Heisenberg group ${\mathcal{H}}$ of $3\times 3$ unipotent matrices and this case was considered in papers [5]—[8] as well as in a more general context, see [1] and [9], say. For example, in [8] the following result was obtained.

Theorem 1

Let $A\subset{\mathbb{R}}$ be a set and

[TABLE]

Then for any $\varepsilon>0$ one has

[TABLE]

Thus formula (1) shows that the products in ${\mathcal{H}}$ are directly connected with the sum–product quantities $AA$ and $A+A$ similar as the products of sets in the affine group. Nevertheless, in a certain sense the affine group is more correlates with the multiplication and the Heisenberg group correlates with the addition, see the discussion of trivial representations in Section 4.

We improve Theorem 1 and, moreover, generalize it for so–called bricks, see Theorem 13 in Section 5.

Theorem 2

Under the same conditions as in Theorem 1 one has

[TABLE]

where $c>0$ is an absolute constant. Moreover, if $A\subseteq{\mathbb{F}}_{p}$, then

[TABLE]

It was conjectured in [8] that, actually, the right exponent in (1) is four and we have obtained $7/2+c$ in ${\mathbb{R}}$.

Using the representation theory and the incidences theory in ${\mathbb{F}}_{p}$, we have found new bounds for products of large subsets from the Heisenberg group as well from the affine group, see Theorem 18 and Corollary 24 below. Also, we improve the dependence of $n$ on $\varepsilon$ as well as the dependence on $|Z|$ in the following result from [6, Theorem 1.3] (see Theorem 16 from Section 5).

Theorem 3

Let $\varepsilon>0$. Then there exists $n_{0}(\varepsilon)$ such that for all $n\geq n_{0}(\varepsilon)$ and any sets $X_{i},Y_{i},Z\subseteq{\mathbb{F}}_{p}$, $i\in[n]$, $X=\prod_{i=1}^{n}X_{i}\subseteq{\mathbb{F}}^{n}_{p}$, $Y=\prod_{i=1}^{n}Y_{i}\subseteq{\mathbb{F}}^{n}_{p}$ if we form

[TABLE]

with

[TABLE]

then $\mathcal{A}^{2}$ contains at least $|\mathcal{A}|/p$ cosets of $[0,0,{\mathbb{F}}_{p}]$.

In [5] it was found an interesting application of products of sets in the Heisenberg group to so–called models of Freiman isomorphisms. It was showed that there is a (nonabelian) group, namely, the Heisenberg group such that any set with the doubling constant less than two does not has any good model, see [25, Section 5.3]. Recall the required definitions and formulate our result.

Let ${\mathbf{G}}_{1}$, ${\mathbf{G}}_{2}$ be groups, $A\subseteq{\mathbf{G}}_{1}$, $B\subseteq{\mathbf{G}}_{2}$ and $s\geq 2$ be a positive integer. A map $\rho:A\to B$ is said to be a Freiman $s$–homomorphism if for all $2s$–tuples $(a_{1},\dots,a_{s},b_{1},\dots,b_{s})\in A^{s}\times B^{s}$ and any signs $\varepsilon_{j}\in\{-1,1\}$, we have

[TABLE]

If moreover $\rho$ is bijective and $\rho^{-1}$ is also a Freiman $s$–homomorphism, then $\rho$ is called a Freiman $s$–isomorphism. In this case $A$ and $B$ are said to be Freiman $s$–isomorphic.

Theorem 4

*Let $n$ be a positive integer and $\varepsilon\in(0,1/6)$ be any real number. Then there is a finite (nonabelian) group $H$ and a set $A_{*}\subset H$ with the following properties:

$1)~{}$ $|A_{*}|>n$, $|A_{*}A_{*}|<2|A_{*}|$;

$2)~{}$ For any $A\subseteq A_{*}$, $|A|\geq|A_{*}|^{1-\varepsilon}$ and any finite group ${\mathbf{G}}$ such that there exists a Freiman $5$–isomorphism from $A$ to ${\mathbf{G}}$, we have $|{\mathbf{G}}|\gg|A|^{1+\frac{1-6\varepsilon}{5}}$.*

It is well–known [4, Proposition 1.2] that in abelian case the situation above is not possible and Theorem 4 shows that the picture changes drastically already in the simplest nonabelian case of a two–step nilpotent group. Previously, in [5] the authors proved an analogue of Theorem 4 for $6$–isomorphisms (our arguments follow their scheme but are slightly simpler). It is easy to see from our proof that, although, the constant $5$ possibly can be improved but it is the limit of the method.

All logarithms are to base $2.$ The signs $\ll$ and $\gg$ are the usual Vinogradov symbols. For a positive integer $n,$ we set $[n]=\{1,\ldots,n\}.$ Having a set $A$, we will write $a\lesssim b$ or $b\gtrsim a$ if $a=O(b\cdot\log^{c}|A|)$, $c>0$.

The author is grateful to Misha Rudnev for useful discussions.

2 Notation

In this paper ${\mathbf{G}}$ is a group with the identity element $e$, ${\mathbb{F}}$ is a field, ${\mathbb{F}}^{*}={\mathbb{F}}\setminus\{0\}$, and $p$ is an odd prime number, ${\mathbb{F}}_{p}={\mathbb{Z}}/p{\mathbb{Z}}$. Also, we use the same letter to denote a set $A\subseteq{\mathbb{F}}$ and its characteristic function $A:{\mathbb{F}}\to\{0,1\}$.

Put $\mathsf{E}^{+}(A,B)$ for the common additive energy of two sets $A,B\subseteq{\mathbb{F}}$ (see, e.g., [25]), that is,

[TABLE]

If $A=B$, then we simply write $\mathsf{E}^{+}(A)$ instead of $\mathsf{E}^{+}(A,A)$ and the quantity $\mathsf{E}^{+}(A)$ is called the additive energy in this case. One can consider $\mathsf{E}^{+}(f)$ for any complex function $f$ as well. More generally, we deal with a higher energy

[TABLE]

Sometimes we use representation function notations like $r_{AB}(x)$ or $r_{A+B}(x)$, which counts the number of ways $x\in{\mathbb{F}}$ can be expressed as a product $ab$ or a sum $a+b$ with $a\in A$, $b\in B$, respectively. Further clearly

[TABLE]

Similarly, one can define $\mathsf{E}^{\times}(A,B)$, $\mathsf{E}^{\times}(A)$, $\mathsf{E}^{\times}(f)$ and so on. In nonabelian setting the energy of a set $A,B\subseteq{\mathbf{G}}$ is (see [21])

[TABLE]

Clearly, $\mathsf{E}(A,B)\leq|A||B|\min\{|A|,|B|\}$ and $\mathsf{E}(A,B)\geq|A||B|$. We write $[A,B]$ for the set of all commutators of $A$ and $B$, namely, $[A,B]=\{aba^{-1}b^{-1}~{}:~{}a\in A,\,b\in B\}$.

We finish this section recalling some notions and simple facts from the representations theory, see, e.g., [18]. For a finite group ${\mathbf{G}}$ let $\widehat{{\mathbf{G}}}$ be the set of all irreducible unitary representations of ${\mathbf{G}}$. It is well–known that size of $\widehat{{\mathbf{G}}}$ coincides with the number of all conjugate classes of ${\mathbf{G}}$. For $\pi\in\widehat{{\mathbf{G}}}$ denote by $d_{\pi}$ the dimension of this representation and we write $\langle\cdot,\cdot\rangle_{HS}$ for the correspondent Hilbert–Schmidt scalar product $\langle A,B\rangle_{HS}:={\rm tr\,}(AB^{*})$, where $A,B$ are any $(d_{\pi}\times d_{\pi})$–matrices. Clearly, $\langle\pi(g)A,\pi(g)B\rangle_{HS}=\langle A,B\rangle_{HS}$. Also, we have $\sum_{\pi\in\widehat{{\mathbf{G}}}}d^{2}_{\pi}=|{\mathbf{G}}|$.

For any $f:{\mathbf{G}}\to{\mathbb{C}}$ and $\pi\in{\mathbf{G}}$ define the matrix $\widehat{f}(\pi)$ which is called the Fourier transform of $f$ at $\pi$ by the formula

[TABLE]

Then the inverse formula takes place

[TABLE]

and the Parseval identity is

[TABLE]

The main property of the Fourier transform is the convolution formula

[TABLE]

where the convolution of two functions $f,g:{\mathbf{G}}\to{\mathbb{C}}$ is defined as

[TABLE]

Finally, it is easy to check that for any matrices $A,B$ one has $\|AB\|_{HS}\leq\|A\|_{op}\|B\|_{HS}$ and $\|A\|_{op}\leq\|A\|_{HS}$, where the operator $l^{2}$–norm $\|A\|_{op}$ is just the absolute value of the maximal eigenvalue of $A$.

3 Preliminaries

Let ${\mathbb{F}}$ be a field. Let $\mathcal{P}\subseteq{\mathbb{F}}\times{\mathbb{F}}$ be a set of points and ${\cal L}$ be a collection of lines in ${\mathbb{F}}\times{\mathbb{F}}$. Having $p\in\mathcal{P}$ and $l\in{\cal L}$, we write

[TABLE]

Put $\mathcal{I}(\mathcal{P},{\cal L})=\sum_{r\in{\cal P},l\in{\cal L}}{\cal I}(p,l)$. We will omit to write the conditions $r\in{\cal P}$ and $l\in{\cal L}$ below.

A trivial upper bound for ${\cal I}({\cal P},{\cal L})$ is

[TABLE]

see, e.g., [25, Section 8]. Further, there is a bound of Vinh [26] (also, see [20, Section 3]) which says that

[TABLE]

where either $\sum_{r}f(r)=0$ or $\sum_{l}g(l)=0$. Finally, a well–known result of Stevens–de Zeeuw gives us an asymptotic formula for the number of points/lines incidences in the case when the set of points forms a Cartesian product, see [23], and also [20].

Theorem 5

Let $A,B\subseteq{\mathbb{F}}_{p}$ be sets, $\mathcal{P}=A\times B$, and $\mathcal{L}$ be a collection of lines in ${\mathbb{F}}^{2}_{p}$. Then

[TABLE]

The proof rests on a well–known points/planes result from [15] (also, see [20], [26]).

Theorem 6

Let $p$ be an odd prime, $\mathcal{P}\subseteq{\mathbb{F}}_{p}^{3}$ be a set of points and $\Pi$ be a collection of planes in ${\mathbb{F}}_{p}^{3}$. Suppose that $|\mathcal{P}|\leq|\Pi|$ and that $k$ is the maximum number of collinear points in $\mathcal{P}$. Then the number of point–planes incidences satisfies

[TABLE]

4 On products of large subsets of the affine group and the Heisenberg group

Let $n\geq 1$ be a positive integer. By ${\mathcal{H}}_{n}$ define the Heisenberg linear group over ${\mathbb{F}}$ consisting of matrices

[TABLE]

For $n=1$ we write ${\mathcal{H}}={\mathcal{H}}_{1}$. The product rule in ${\mathcal{H}}_{n}$ is

[TABLE]

where $xy^{\prime}$ is the scalar product of vectors $\vec{x}$ and $\vec{y^{\prime}}$. Also, one has

[TABLE]

Clearly, $|{\mathcal{H}}_{n}|=|{\mathbb{F}}|^{2n+1}$ and there are $|{\mathbb{F}}|^{2n}+|{\mathbb{F}}|-1$ conjugate classes of the form $[x,y,0]$, $(x,y)\neq 0$ and $[0,0,z]$, $z\in{\mathbb{F}}$. For any $a,a^{\prime}\in{\mathcal{H}}_{n}$, $a=[x,y,z]$, $a^{\prime}=[x^{\prime},y^{\prime},z^{\prime}]$, their commutator equals $[a;a^{\prime}]=[0,0,xy^{\prime}-yx^{\prime}]$. Thus the centre of ${\mathcal{H}}_{n}$ is $[0,0,z]$, $z\in{\mathbb{F}}$ and hence ${\mathcal{H}}_{n}$ is a two–step nilpotent group. Given $[x_{0},y_{0},z_{0}]\in{\mathcal{H}}_{n}$, we see that the centralizer $\mathrm{C}([x_{0},y_{0},z_{0}])=\{[x,y,z]~{}:~{}xy_{0}=x_{0}y\}$. The Heisenberg group ${\mathcal{H}}_{n}$ acts on ${\mathbb{F}}^{n}\times{\mathbb{F}}^{n}$ as

[TABLE]

and hence ${\rm Stab\,}((x,y))=\{[a,b,0]~{}:~{}ay+b=0\}$. Further the structure of $\widehat{{\mathcal{H}}}_{n}$ is well–known, see, e.g., [17]. There are $|{\mathbb{F}}|^{2n}$ one–dimensional representations which correspond to additive characters for $x,y$, see the group low (13) and there is a unique nontrivial representation $\pi$ of dimension $|{\mathbb{F}}|^{n}$. Thus formula (6) has the following form

[TABLE]

where $\delta_{f}(x,y)=\sum_{z}f([x,y,z])$. Let us describe the representation $\pi$ in details in the case $n=1$, see, e.g., [17]. Let $\zeta=e^{2\pi i/p}$ and $\mathcal{D}=diag(1,\zeta,\dots,\zeta^{p-1})$ and

[TABLE]

be $(p\times p)$ matrix. Then $\pi([x,y,z]):=\zeta^{z+y}\mathcal{D}^{y}W^{x}_{\zeta}$. The fact that $\pi$ is a representation follows from an easy checkable commutative identity

[TABLE]

Thus there is just one nontrivial representation $\pi$ and a similar situation takes place in the case of the affine group ${\rm Aff\,}({\mathbb{F}})$, see below.

Now we obtain a lemma on products of sets in ${\mathcal{H}}_{n}$. A similar result was obtained in [7, Propositions 3–6 and Theorem 1] but for a special family of sets which are called semi–bricks. Given a set $A\subseteq{\mathcal{H}}$ we write $K^{-1}(A):=|A|^{-1}\max_{x,y}\delta_{A}(x,y)$. Hence from the definition of the quantity $K=K(A)$ one has that for any $x,y\in{\mathbb{F}}$ the following holds $\delta_{A}(x,y)\leq|A|/K$.

Theorem 7

*Let $A,B\subseteq{\mathcal{H}}$, $|A||B|>p^{5}$. Then $[A,B]$ contains $[0,0,{\mathbb{F}}_{p}]$.

Further if $A\subseteq{\mathcal{H}}_{n}$, then for any $k\geq 2$ and any signs $\varepsilon_{j}\in\{-1,1\}$ with $\sum_{j=1}^{2k}\varepsilon_{j}=0$ the product $\prod_{j=1}^{2k}A^{\varepsilon_{j}}$ contains $[0,0,{\mathbb{F}}_{p}]$, provided*

[TABLE]

Finally, for $K=K(A)$ and $k\geq 2$, we have

[TABLE]

P r o o f.  We know that for any $a,b\in{\mathcal{H}}_{n}$, $a=[x,y,z]$, $b=[x^{\prime},y^{\prime},z^{\prime}]$ their commutator equals $[a;b]=[0,0,xy^{\prime}-yx^{\prime}]$. Hence for any $\lambda\neq 0$ we must solve the equation $xy^{\prime}-yx^{\prime}=\lambda$, where points $(x,y)$ and $(x^{\prime},y^{\prime})$ are counted with the weights equal $\delta_{A}$ and $\delta_{B}$. Using Theorem of Vinh (10), we see that the number solutions to this equation is at least

[TABLE]

because of our assumption $|A||B|>p^{5}$ and a trivial estimate $\|\delta_{A}\|_{2}\|\delta_{B}\|_{2}\leq p(|A||B|)^{1/2}$.

To prove the second part of the theorem take any $z_{*}:=[0,0,z]\in[0,0,{\mathbb{F}}_{p}]$ and write $S$ for the convolution of $\prod_{j=1}^{2k}A^{\varepsilon_{j}}$. Then by (8), we have $\widehat{S}(\pi)=\prod_{j=1}^{2k}{\mathcal{C}}^{\varepsilon_{j}}\widehat{A}(\pi)$, where ${\mathcal{C}}$ is the conjugation operator. Using (8) and the fact that all one–dimensional representations equal $1$ on $[0,0,{\mathbb{F}}_{p}]$, we obtain

[TABLE]

[TABLE]

Here we have used the Parseval identity (7). On the other hand, applying the Parseval formula again, we get

[TABLE]

and hence

[TABLE]

Substituting the last bound into (19), we derive

[TABLE]

as required.

To obtain (18) we use the calculations above and, applying definition of $\mathsf{T}_{k}(A)$ from (4), we obtain

[TABLE]

Using the Cauchy–Schwarz inequality, we derive $|A|^{2k}\leq\mathsf{T}_{k}(A)|A^{k}|$ and hence we complete the proof. $\hfill\Box$

Remark 8

A variant of the second part of the lemma above can be obtained for products of different sets and we leave it to the interested reader. Clearly, a lower bound for size of $A$ such that $A^{n}$ contains $[0,0,{\mathbb{F}}_{p}]$ is $\Omega_{n}(p^{n+1})$ even in the symmetric case, indeed just consider all matrices $[0,{\mathbb{F}}^{n}_{p},P]$, where $|P|<p/(2n)$ is an arithmetic progression.

Now we need a result from [5, Lemma 2].

Lemma 9

Let ${\mathbf{G}}$ be a group and $X$ be a maximal subset of ${\mathbf{G}}$ such that

[TABLE]

Then $XX=X$.

The first part of Theorem 18 combined with Lemma 9 imply the following consequence.

Theorem 10

*Let $n$ be a positive integer and $\varepsilon\in(0,1/6)$ be any real number. Then there is a finite (nonabelian) group $H$ and a set $A_{*}\subset H$ with the following properties:

$1)~{}$ $|A_{*}|>n$, $|A_{*}A_{*}|<2|A_{*}|$;

$2)~{}$ For any $A\subseteq A_{*}$, $|A|\geq|A_{*}|^{1-\varepsilon}$ and any finite group ${\mathbf{G}}$ such that there exists a Freiman $5$–isomorphism from $A$ to ${\mathbf{G}}$, we have $|{\mathbf{G}}|\gg|A|^{1+\frac{1-6\varepsilon}{5}}$.*

P r o o f.  The argument follows the scheme of the proof from [5]. Let

[TABLE]

and we will choose $\alpha\in(0,1)$ later. Clearly, $|A_{*}A_{*}|\leq 2|A_{*}|-p^{2}<2|A_{*}|$. Take any $A\subseteq A_{*}$, $|A|\geq|A_{*}|^{1-\varepsilon}$ and let $\rho$ be a Freiman $5$–isomorphism from $A$ to a group ${\mathbf{G}}$. We can assume that ${\mathbf{G}}=\langle\rho(A)\rangle$ and using Lemma 9, we derive that ${\mathbf{G}}$ is a two–step nilpotent group. If

[TABLE]

then by Theorem 18 the set $B:=[A,A]\subseteq AAA^{-1}A^{-1}$ contains $[0,0,{\mathbb{F}}_{p}]$. One satisfies the last condition taking $\alpha=\frac{1+4\varepsilon}{2-2\varepsilon}$. We write $g_{z}\in{\mathbf{G}}$ for $g_{z}=\rho([0,0,z])$, $z\in{\mathbb{F}}_{p}$. Further by the average arguments one can find $u,v\in{\mathbb{F}}_{p}$ and a set $Z\subseteq{\mathbb{F}}_{p}$ such that $[u,v,Z]\subseteq A$ and for $p$ large enough the following holds $|Z|\geq|A|/4p^{1+\alpha}>1$. Taking two distinct elements $[u,v,i]$, $[u,v,j]\in A$ and putting $h_{k}=\rho([u,v,k])$, where $k=i,j$, we form $g_{i-j}:=h_{j}^{-1}h_{i}\in{\mathbf{G}}$, $g_{i-j}\neq e$. Finally, $\rho(B)$ contains $\rho([0,0,{\mathbb{F}}_{p}])$, hence $g_{i-j}\in\rho(B)$ and one can check by induction (see [5]) that for any $l\geq 1$ the following holds $g_{l(i-j)}=g^{l}_{i-j}$. In particular, the order of $g_{i-j}$ in ${\mathbf{G}}$ is $p$. Consider Sylow $p$–subgroup of ${\mathbf{G}}$ which we denote by ${\mathbf{G}}_{p}$. Suppose that ${\mathbf{G}}_{p}$ is abelian. We know that $[a;a^{\prime}]=[0,0,xy^{\prime}-yx^{\prime}]$ for any $a,a^{\prime}\in A$ and since $\rho$ is $5$–isomorphism and hence $4$–isomorphism, it follows that $xy^{\prime}-yx^{\prime}=0$ on $A$, whence $|A|\leq p^{2}$ and this is a contradiction since $|A|>p^{(2+\alpha)(1-\varepsilon)}$ and this contradicts with our choice of the parameter $\alpha$ (see details in [5]). Otherwise, ${\mathbf{G}}_{p}$ is nonabelian and in view of (20) and our choice of $\alpha$, we obtain

[TABLE]

as required. $\hfill\Box$

It is easy to see from the proof that, although, possibly, the constant $5$ can be improved but it is the limit of the method.

Now consider the group of invertible affine transformations ${\rm Aff\,}({\mathbb{F}})$ of a field ${\mathbb{F}}$, i.e., maps of the form $x\to ax+b$, $a\in{\mathbb{F}}^{*},b\in{\mathbb{F}}$ or, in other words, the set of matrices

[TABLE]

Here we associate with such a matrix the vector $(a,b)$. Then ${\rm Aff\,}({\mathbb{F}})$ is a semi–product ${\mathbb{F}}_{p}^{*}\ltimes{\mathbb{F}}_{p}$ with the multiplication $(a,b)\cdot(c,d)=(ac,ad+b)$. Clearly, ${\rm Aff\,}({\mathbb{F}})$ acts on ${\mathbb{F}}$. For any $a,a^{\prime}\in{\rm Aff\,}({\mathbb{F}})$, $a=(x,y)$, $a^{\prime}=(x^{\prime},y^{\prime})$, their commutator equals $[a,a^{\prime}]=(1,y(1-x^{\prime})-y^{\prime}(1-x))$. The group ${\rm Aff\,}({\mathbb{F}})$ contains the standard unipotent subgroup $U=\{(1,a)~{}:~{}a\in{\mathbb{F}}\}$ as well as the standard dilation subgroup $T=\{(a,0)~{}:~{}a\in{\mathbb{F}}^{*}\}$. The centralizer $\mathrm{C}(I)$ of $I$ is ${\rm Aff\,}({\mathbb{F}})$, further, if $g=(x,y)\in U\setminus\{I\}$, then $\mathrm{C}(g)=U$ and otherwise $\mathrm{C}(g)={\rm Stab\,}(y(1-x)^{-1})$, where ${\rm Stab\,}(x_{0})=\{(a,x_{0}(1-a))~{}:~{}a\in{\mathbb{F}}^{*}\}$. The subgroups $U$ and $T$ are maximal abelian subgroups of ${\rm Aff\,}({\mathbb{F}})$.

There are $(|{\mathbb{F}}|-1)$ one–dimensional representations which correspond to multiplicative characters of ${\mathbb{F}}^{*}$ and because there exist precisely $|{\mathbb{F}}|$ conjugate classes in ${\rm Aff\,}({\mathbb{F}})$ we see that there is one more nontrivial representation $\pi$ of dimension $|{\mathbb{F}}|-1$. We have an analogue of formula (15)

[TABLE]

where $\delta_{f}(x)=\sum_{y}f((x,y))$. As above let us describe the representation $\pi$ in details, see, e.g., [2]. We define $\mathcal{D}=diag(1,\zeta^{\omega},\dots,\zeta^{\omega^{p-2}})$, where $\omega$ is any primitive root in ${\mathbb{F}}_{p}^{*}$. Then $\pi((x,y)):=\mathcal{D}^{y}W^{{\rm ind}(x)}_{1}$ (now $W_{1}$ is $(p-1)\times(p-1)$ matrix). An analogue of identity (16) is

[TABLE]

Hence as in the case of the Heisenberg group there is just one nontrivial representation $\pi$ of large dimension and thanks to this similarity we can consider these two groups together. Underline it one more time that the trivial representations of ${\mathcal{H}}$ correspond to additive characters but the trivial representations of ${\rm Aff\,}$ correspond to multiplicative ones.

Put $K^{-1}(A):=|A|^{-1}\max_{x}\delta_{A}(x)$. Using the same method as in the proof of Theorem 18, one has

Corollary 11

*Let $A\subseteq{\rm Aff\,}({\mathbb{F}}_{p})$, $|A|>p^{3/2}$. Then $[A,A]$ contains $(1,{\mathbb{F}}_{p})$.

Further for any $k\geq 2$ and any signs $\varepsilon_{j}\in\{-1,1\}$ with $\sum_{j=1}^{2k}\varepsilon_{j}=0$ the product $\prod_{j=1}^{2k}A^{\varepsilon_{j}}$ contains $(1,{\mathbb{F}}_{p})$, provided*

[TABLE]

For $K=K(A)$ and $k\geq 2$, we have

[TABLE]

As in Remark 8 a lower bound for size of $A$ such that $A^{n}$ contains $(1,{\mathbb{F}}_{p})$ is $\Omega(p)$ because one can consider the set of all matrices $({\mathbb{F}}_{p},0)$ as an example.

Let us demonstrate just one particular usage of Corollary 24.

Example 12

Let $A=\{(a,b)~{}:~{}a\in{\mathbb{F}}^{*}_{p},\,b\in{\mathbb{F}}_{p}\}\subseteq{\rm Aff\,}({\mathbb{F}}_{p})$ and $|A|>p^{3/4}$. Then considering $A^{-1}AA^{-1}A$, we see that for any $\lambda\in{\mathbb{F}}_{p}$ there are $a_{i},b_{i},c_{i}\in A$ such that

[TABLE]

5 On products of bricks in the Heisenberg group and in the affine group

Now let us obtain an upper bound for the energy of bricks in ${\mathcal{H}}$, see the definition in Theorem 13 below. In particular, it gives a lower bound for size of the product set of such sets.

Theorem 13

Let $\mathcal{A}=\{[x,y,z]~{}:~{}x\in X,\,y\in Y,\,z\in Z\}\subseteq{\mathcal{H}}$ be a set. Put $M=\max\{|X|,|Y|\}$. Then

[TABLE]

where $\mathcal{E}$ is

[TABLE]

P r o o f.  The energy $\mathsf{E}(\mathcal{A})$ equals the number of the solutions to the system

[TABLE]

where $x,x^{\prime},x_{*},x^{\prime}_{*}\in X$, $y,y^{\prime},y_{*},y^{\prime}_{*}\in Y$, $z,z^{\prime},z_{*},z^{\prime}_{*}\in Z$. First of all we consider solutions to (26) with all possible $z,z^{\prime},z_{*},z^{\prime}_{*}\in Z$ such that $z+z_{*}\neq z^{\prime}+z^{\prime}_{*}$. Denote by $\sigma_{1}$ the correspondent number of the solutions. Then the last equation of our system (26) determines a line such that $(x,x^{\prime})\in X\times X$ and $(y_{*},y^{\prime}_{*})\in Y\times Y$ are counted with the weights $r_{X-X}(x-x^{\prime})$ and $r_{Y-Y}(y_{*}-y^{\prime}_{*})$, correspondingly. Clearly, such weights do not exceed $|X|$ and $|Y|$, respectively. Moreover, $\sum_{x,x^{\prime}\in X}r_{X-X}(x-x^{\prime})=\mathsf{E}^{+}(X)$ and similar $\sum_{y,y^{\prime}\in Y}r_{Y-Y}(y-y^{\prime})=\mathsf{E}^{+}(Y)$ . Using the pigeonhole principle and applying Theorem 11, we find a number $0<\Delta\leq|X|$ and a set of lines $L\subseteq X\times X$, $\Delta|L|\leq\mathsf{E}^{+}(X)$ such that

[TABLE]

[TABLE]

Let us give another estimate for $\sigma_{1}$. Now we crudely bound $r_{X-X}(x-x^{\prime})$ and $r_{Y-Y}(y_{*}-y^{\prime}_{*})$ as $|X|$ and $|Y|$, respectively, but treat our equation $z+z_{*}+xy_{*}=z^{\prime}+z^{\prime}_{*}+x^{\prime}y^{\prime}_{*}$ as $s+xy_{*}=s^{\prime}+x^{\prime}y^{\prime}_{*}$, where $s,s^{\prime}$ are counted with weights $r_{Z+Z}(s)$, $r_{Z+Z}(s^{\prime})$. Applying Theorem 12 and using the same calculations as above, we obtain

[TABLE]

as required.

Now consider the remaining case when $z+z_{*}=z^{\prime}+z^{\prime}_{*}$ and denote the rest by $\sigma_{2}/\mathsf{E}^{+}(Z)$. One can check that zero solutions in the remaining variables $x$, $x^{\prime}$, $y^{\prime}_{*}$, $y_{*}$ as well as solutions with $\alpha:=x/x^{\prime}=y^{\prime}_{*}/y_{*}=1$ coins at most

[TABLE]

in $\sigma_{2}$. Thus suppose that $\alpha\neq 1$ and all variables $x$, $x^{\prime}$, $y^{\prime}_{*}$, $y_{*}$ do not vanish. We have

[TABLE]

In particular, $\frac{x^{\prime}}{y_{*}}=\frac{x^{\prime}_{*}-x_{*}}{y-y^{\prime}}$ and if we determine all variables $x^{\prime},x_{*},x^{\prime}_{*},y,y^{\prime},y_{*}$ from the last equation, then from (30), we know $\alpha$ and hence recalling $\alpha=x/x^{\prime}=y^{\prime}_{*}/y_{*}$, we find the remaining variables $x,y^{\prime}_{*}$. Hence

[TABLE]

Using Theorem 12, we get

[TABLE]

Combining the last estimate, bounds (27), (28) and (29), we obtain the required result. $\hfill\Box$

For example, if $|X|=|Y|=|Z|\leq p^{2/3}$, then the result above gives us $\mathsf{E}(\mathcal{A})\lesssim|\mathcal{A}|^{3-1/6}$.

Now if $Z=\{0\}$, then we do not need to consider the first case in the proof of Theorem 13, hence $\mathcal{E}=0$ and whence, we obtain a consequence which is better than [8, Theorem 2.4].

Corollary 14

Let $\mathcal{A}=\{[x,y,0]~{}:~{}x,y\in A\}\subseteq{\mathcal{H}}$ be a set. Then

[TABLE]

Remark 15

It was proved in [16] that the quantity from (31) can be estimated better for real sets $A\subset{\mathbb{R}}$, namely, as $O(|A|^{9/2-c})$, where $c>0$ is an absolute constant. Hence in ${\mathbb{R}}$ lower bound (32) in Corollary 32 is even better.

We say that two series of sets $X_{i}\subseteq{\mathbb{F}}_{p}$, $Y_{i}\subseteq{\mathbb{F}}_{p}$ have comparable sizes if for all $i,j\in[n]$ the following holds $|X_{i}|\ll|X_{j}|$, $|Y_{i}|\ll|Y_{j}|$. In this case put $\mathcal{X}=\max_{i\in[n]}|X_{i}|$, $\mathcal{Y}=\max_{i\in[n]}|Y_{i}|$.

Now we are ready to improve Theorem 3 from the Introduction in the situation when $X_{i},Y_{i}$ have comparable sizes. It is easy to show that in our result $\varepsilon(n)=\lambda^{n}$ for a certain $\lambda<1$ but in Theorem 3 it is just $\varepsilon(n)=O(1/n)$. Also, the dependence on $|Z|$ in Theorem 16 is better. Finally, we remark that of course the lower bound $|\mathcal{A}|/p$ for the number of cosets is optimal.

Theorem 16

Let $n\geq 2$ be an even number, and $X_{i},Y_{i},Z\subseteq{\mathbb{F}}_{p}$, $i\in[n]$, $X=\prod_{i=1}^{n}X_{i}\subseteq{\mathbb{F}}^{n}_{p}$, $Y=\prod_{i=1}^{n}Y_{i}\subseteq{\mathbb{F}}^{n}_{p}$,

[TABLE]

be sets and $X_{i}$, $Y_{i}$ have comparable sizes. If $|Z|\leq\mathcal{X}\mathcal{Y}$, $\mathcal{X}\leq|Z|\mathcal{Y}$, $\mathcal{Y}\leq|Z|\mathcal{X}$ and

[TABLE]

then $\mathcal{A}^{2}$ contains at least $|\mathcal{A}|/p$ cosets of $[0,0,{\mathbb{F}}_{p}]$.

P r o o f.  It is enough to prove that $\mathcal{A}^{2}$ contains $[0,0,{\mathbb{F}}_{p}]$, provided

[TABLE]

because then (33) follows by arguments from [6, Theorem 1.3]. Indeed, if we replace $X_{i}$, $Y_{i}$ by $\tilde{X}_{i}:=X_{i}\cap(a_{i}-X_{i})$, $\tilde{Y}_{i}:=Y_{i}\cap(b_{i}-Y_{i})$ for some $a_{i},b_{i}$ and consider the correspondent set $\tilde{\mathcal{A}}$, then by the group low (13) the inclusion $[0,0,{\mathbb{F}}_{p}]\subseteq\tilde{\mathcal{A}}^{2}$ implies $[\vec{a},\vec{b},{\mathbb{F}}_{p}]\subseteq\mathcal{A}^{2}$, where $\vec{a}=(a_{1},\dots,a_{n})$, $\vec{b}=(b_{1},\dots,b_{n})$. Further notice that the set $\Omega_{i}=\{a~{}:~{}|X_{i}\cap(a-X_{i})|\geq\zeta|X_{i}|^{2}/p\}$ has size $|\Omega_{i}|\geq(1-\zeta)|X_{i}|$ and hence taking $\zeta$ such that $(1-\zeta)^{2n}\geq 1/2$ we can find at least

[TABLE]

vectors $\vec{a}$, $\vec{b}$ with $|\tilde{X}_{i}|\geq\zeta|X_{i}|^{2}/p$, $|\tilde{Y}_{i}|\geq\zeta|Y_{i}|^{2}/p$. To get (35) we have used the fact that $|Z|<p/2$ because otherwise Theorem 16 is trivial. Substitution $\tilde{X}_{i}$, $\tilde{Y}_{i}$ into (34) gives the desired condition (33).

Now let us obtain (34). Take $[x,y,z],[x^{\prime},y^{\prime},z^{\prime}]\in\mathcal{A}$ and by the group low (13) we need to solve the equation

[TABLE]

for any $\lambda$. We consider even $n$ only (recall that we assume that $n\geq 2$) and denote by $\sigma_{n/2}$ the number of the solutions to the last equation. Almost repeating the proof of [20, Theorem 32] (also, see [20, Remark 33]), one obtains an asymptotic formula for $\sigma_{k}$, namely,

[TABLE]

Indeed, by Theorem 12 we know (thanks to $|Z|\leq\mathcal{X}\mathcal{Y}$, $\mathcal{X}\leq|Z|\mathcal{Y}$, $\mathcal{Y}\leq|Z|\mathcal{X}$) that

[TABLE]

and that the recurrent formula for the error term $\mathcal{E}_{k}$ in the right–hand side of (36) is

[TABLE]

Again we need to use our conditions $|Z|\leq\mathcal{X}\mathcal{Y}$, $\mathcal{X}\leq|Z|\mathcal{Y}$, $\mathcal{Y}\leq|Z|\mathcal{X}$ and induction similar to the proof of [20, Theorem 32]. Thus asymptotic formula (36) takes place and $\sigma_{k}$ is positive if

[TABLE]

This completes the proof. $\hfill\Box$

Let us compare condition (17) of Theorem 18 (condition (23) of Corollary 24) and Theorem 16 namely, formula (34) from the proof. Theorem 18 concerns general sets but condition (34) is exponentially better than (17). For concrete families of sets one can prove similar exponentially small bounds. Consider, for example, a brick $\mathcal{A}=\{(a,b)~{}:~{}a,b\in A\}\subseteq{\rm Aff\,}({\mathbb{F}}_{p})$, $A\subseteq{\mathbb{F}}^{*}_{p}$ and give the sketch of the proof of the existence of this decay (see details in [20, Remark 34] and in [16, Theorem 11]). Put $\mathcal{A}^{n}=\{(a_{n},b_{n})\}$ and by the group low we know that $a_{n+1}=a_{n}a$, $b_{n+1}=a_{n}b+b_{n}$, where $a,b\in A$. Using the last recursive formula and the arguments as in [20, Theorem 32] to solve the equation $a_{n}b+b_{n}=a^{\prime}_{n}b^{\prime}+b^{\prime}_{n}$, we obtain in ${\mathbb{R}}$ (but similar in ${\mathbb{F}}_{p}$) that for any $\mathcal{B}$ from the affine group one has $|\mathcal{B}\mathcal{A}|\gg|A|^{3/2}|\mathcal{B}|^{1/2}$ and this implies the exponential decay.

6 Concluding remarks

In this section we discuss some further connections between the sum–product phenomenon and growth in the Heisenberg group.

In Theorem 13 we have deal with the term $\sigma_{2}$. It is easy to see that this quantity is just $\sum_{\lambda,\mu}\mathsf{E}^{\times}(A^{+}_{\lambda},A^{+}_{\mu})\,,$ where $A^{+}_{\lambda}=A\cap(\lambda-A)$. Hence we have estimated this expression as well. In a dual way one can consider $\sum_{\lambda,\mu}\mathsf{E}^{+}(A^{\times}_{\lambda},A^{\times}_{\mu})\,,$ where $A^{\times}_{\lambda}=A\cap\lambda A^{-1}$ or, similarly, $A^{\times}_{\lambda}=A\cap\lambda A$. Then we have the correspondent analogue of system (26), namely,

[TABLE]

It gives $b_{1}b/b^{\prime}-b_{1}=a^{\prime}a^{\prime}_{1}/a_{1}-a^{\prime}$ (and the remaining variables $a,b^{\prime}_{1}$ can be find uniquely) and hence again this can be bounded as $|A|^{6}/p+O(|A|^{9/2})$ in ${\mathbb{F}}_{p}$ and as $O(|A|^{9/2-c})$ in ${\mathbb{R}}$, where $c>0$ is an absolute constant, see [16].

In a similar way, one can consider the problem of estimating the quantities

[TABLE]

The first one naturally appears in sum–product questions in ${\mathbb{R}}$ which are connected with Solymosi’s argument [22], see, e.g., [10]. As in Theorem 13, we see that the first sum equals the number solutions to the system

[TABLE]

hence as above $\frac{a^{\prime}}{b_{1}}=\frac{a^{\prime}_{1}-a_{1}}{b-b^{\prime}}$ and after some calculations we arrive to

[TABLE]

Now we can estimate the number solutions to the last equation rather roughly. Indeed, if we fix a variable, say, $b_{1}$, then relatively to $a_{1},b^{\prime}$ we have an equation of a line. Hence the Szemerédi–Trotter Theorem [24] gives us $\sum_{\lambda}\mathsf{E}^{\times}(A^{+}_{\lambda})\ll|A|^{11/3}$ and similar in ${\mathbb{F}}_{p}$ via Theorem 11. One can estimate the number solutions to (38) further via the Cauchy–Schwarz and different energies.

As for the dual question, it is easy to see that $\sum_{\lambda}\mathsf{E}^{+}(A\cap\lambda A^{-1})\leq|A|\mathsf{E}^{+}(A^{-1})$ and $\sum_{\lambda}\mathsf{E}^{+}(A\cap\lambda A)\leq|A|\mathsf{E}^{+}(A)$ because the map $(x,y,z,w,\lambda)_{x+y=z+w}\to\lambda^{-1}(x,y,z,w)_{x+y=z+w}$ has at most $|A|$ preimages. Thus in this case nothing interesting happens and one needs a deeper technique to estimate the sum.

Problem. Estimate the sum–product quantities (37) in ${\mathbb{R}}$ and in ${\mathbb{F}}_{p}$ (for small $A$). We suppose that the correct bound is $O(|A|^{3+\varepsilon})$ for an arbitrary $\varepsilon>0$.