Expansion for the product of matrices in groups

TL;DR

This paper establishes lower bounds on the size of product sets of matrices in specific groups, extending known results to matrix groups like SL_2 over finite fields and the Heisenberg group.

Contribution

It provides new lower bounds for matrix product sets in groups, generalizing previous scalar and cube expansion results to matrix groups.

Findings

Proved lower bounds for matrix product sets in SL_2(𝔽_p).

Extended expansion results to the Heisenberg group.

Generalized scalar expansion results to matrix groups.

Abstract

In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(\mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyv\'{a}ri and Hennecart.

Equations

$$R(A):=\left\{\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\in S{L}_2({\mathbb{F}}_p):a_{11},a_{12},a_{21}\in A\right\}.$$

$$|R(A)\cdot R(A)|\gg p^3.$$

$$|R(A)\cdot R(A)|\gg |A{|}^{\frac{7}{2}+\frac{1}{12}}.$$

$$[\mathbf{x},\mathbf{y},z]:=\left[\begin{array}{ccc}1& \mathbf{x}& z\\ 0& I_n& {\mathbf{y}}^{\mathbf{t}}\\ 0& 0& 1\end{array}\right]$$

$$[A^n,B^n,C]:=\{[\mathbf{x},\mathbf{y},z]:\mathbf{x}\in A^n,\mathbf{y}\in B^n,z\in C\}.$$

$$|[A^n,B^n,C]|>|{\mathbf{H}}_n(({\mathbb{F}}_p)){|}^{3/4+\epsilon },$$

$$|[A,A,0]\cdot [A,A,0]|\gg \min \left\{p^{1/2}|A{|}^{5/2},p^{-1/2}|A{|}^4\right\}.$$

$$|[A,A,0]\cdot [A,A,0]|\gg |A{|}^{\frac{7}{2}}.$$

$$|[A^2,A^2,0{]}^2|\gtrsim |A{|}^{\frac{11}{2}+\frac{25}{262}}.$$

$$|[A^2,A^2,A{]}^2|\gtrsim |A{|}^{\frac{11}{2}+\frac{23}{90}}.$$

$$|X+A\cdot A|\gg \min \left\{|X{|}^{1/2}|A|,p\right\}.$$

$$a_1{a}_2+a_3{a}_4=a_1^{\prime }{a}_2^{\prime }+a_3^{\prime }{a}_4^{\prime }$$

$$a_1{a}_2+\lambda a_3=a_1^{\prime }{a}_2^{\prime }+\beta a_3^{\prime }$$

$$a_1{a}_2+a_3{a}_4=a_1^{\prime }{a}_2^{\prime }+a_3^{\prime }{a}_4^{\prime }$$

$$I(A\times B,L)\ll |A{|}^{3/4}|B{|}^{1/2}|L{|}^{3/4}+|L|.$$

$$|A\cap (B+x)|\ll |A{|}^{-1/2}|AB{|}^{5/4},$$

$$|A\cap (B+x)|\ll \frac{1}{|A||B|}\left|\left\{(u,u_{\ast },a,b)\in AB\times AB\times A\times B:u{b}^{-1}-u_{\ast }{a}^{-1}=x\right\}\right|.$$

$$I(A^{-1}\times AB,L)\ll |A{|}^{3/2}|AB{|}^{5/4},$$

$$|A\pm A{|}^{18}|AA{|}^9\gtrsim |A{|}^{32}.$$

$$M_1:=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& \frac{1+a_{12}{a}_{21}}{a_{11}}\end{array}\right),M_2:=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& \frac{1+b_{12}{b}_{21}}{b_{11}}\end{array}\right).$$

$$M_1\cdot M_2=\left(\begin{array}{cc}t& \alpha \\ \beta & \frac{1+\alpha \beta }{t}\end{array}\right),$$

$$a_{11}{b}_{11}+a_{12}{b}_{21}=t,\frac{b_{12}t}{b_{11}}+\frac{a_{12}}{b_{11}}=\alpha ,\frac{a_{21}t}{a_{11}}+\frac{b_{21}}{a_{11}}=\beta .$$

$$|AA+AA|\gg \min \{|A{|}^{\frac{3}{2}+\frac{ϵ}{2}},p\}=|A{|}^{\frac{3}{2}+\frac{ϵ}{2}},$$

$$|A|\le p^{\frac{2}{3+ϵ}}.$$

$$|R(A)\cdot R(A)|\gg |AA+AA||A{|}^2\gg |A{|}^{\frac{7}{2}+\frac{ϵ}{2}},$$

$$\sum _{\alpha ,\beta }\nu (0,\alpha ,\beta )\le |A{|}^5.$$

$$(|A{|}^6-|A{|}^5{)}^2\le {\left(\sum _{t\ne 0,\alpha ,\beta }\nu (t,\alpha ,\beta )\right)}^2\le |R(A)\cdot R(A)|\sum _{t\ne 0,\alpha ,\beta }{\nu }^2(t,\alpha ,\beta ).$$

$$|R(A)\cdot R(A)|\gg \frac{|A{|}^{12}}{T},$$

$$T\ll |A{|}^{8+\frac{5ϵ}{2}}.$$

$$a_{11}{b}_{11}+a_{12}{b}_{21}$$

Full text

1 Introduction

Let $\mathbb{F}_{p}$ be a prime field. We denote by $SL_{2}(\mathbb{F}_{p})$ the set of $2\times 2$ matrices with determinant one over $\mathbb{F}_{p}$ . Given $A\subset\mathbb{F}_{p}$, we define

[TABLE]

It was proved by Chapman and Iosevich [1] by Fourier analytic methods that if $|A|\gg p^{5/6}$ then

[TABLE]

Throughout this paper the notation $U\ll V$ means $U\leq cV$ for some absolute constant $c>0$, and $U\gtrsim V$ means $U\gg(\log U)^{-c}V$ for some absolute constant $c>0$. It has been extensively studied about the size of the products of $R(A)$. In particular, the breakthrough work of H. A. Helfgott [2] asserts that if $E$ is a subset of $SL_{2}(\mathbb{F}_{p})$ and is not contained in any proper subgroup with $|E|<p^{3-\delta}$, then $|E\cdot E\cdot E|>c|E|^{1+\epsilon}$ for some $\epsilon=\epsilon(\delta)>0$. The result mentioned above by Chapman and Iosevich is to give a quantitative estimate when the size of the set $A$ is large. However it is considered to be a difficult problem to obtain some quantitative estimate for the same problem when the size of the set $A$ is not large. It is basically because the Fourier analytic methods are effective only when the size of the set $A$ is large. In this paper, we address the case of small sets, and give a lower bound on the size of $R(A)\cdot R(A)$. Our first result is as follows.

Theorem 1.1**.**

Let $A\subset\mathbb{F}_{p}.$ If $|A|\leq cp^{\frac{12}{19}}$ for some small constant $c>0$, then

[TABLE]

Let $\mathbb{F}_{p}$ be a prime field. For an integer $n\geq 1$, the Heisenberg group of degree $n$, denoted by $\mathbf{H}_{n}(\mathbb{F}_{p}),$ is defined by a set of the following matrices:

[TABLE]

where $\mathbf{x},\mathbf{y}\in\mathbb{F}_{p}^{n}$, $z\in\mathbb{F}_{p}$, $\mathbf{y^{t}}$ denotes the column vector of $\mathbf{y}$, and $I_{n}$ is the $n\times n$ identity matrix. For $A,B,C\subset\mathbb{F}_{p}$, we define

[TABLE]

A similar question in the setting of the Heisenberg group over prime fields has been recently investigated by Hegyvári and Hennecart in [4], namely, they proved the following theorem.

Theorem 1.2** (Hegyvári-Hennecart, [4]).**

For every $\varepsilon>0,$ there exists a positive integer $n_{0}=n_{0}(\epsilon)$ such that if $n\geq n_{0}$, and $[A^{n},B^{n},C]\subseteq\mathbf{H}_{n}(\mathbb{F}_{p})$ with

[TABLE]

then there exists a non-trivial subgroup $G$ of $\mathbf{H}_{n}(\mathbb{F}_{p})$ such that $[A^{n},B^{n},C]\cdot[A^{n},B^{n},C]$ contains at least $|[A^{n},B^{n},C]|/p$ cosets of $G$.

In a very recent paper, using results on sum-product estimates, Hegyvári and Hennecart [3] established some results in the case $n=1$. In particular, they proved that if $A\subset\mathbb{F}_{p}$ with $|A|\geq p^{1/2}$, then

[TABLE]

In the case when $|A|\leq p^{2/3}$, they also showed that

[TABLE]

In this paper, we also extend this result to the setting of Heisenberg group of degree two. For simplicity, we write $[A^{2},A^{2},0]^{2}$ and $[A^{2},A^{2},A]^{2}$ for the products $[A^{2},A^{2},0]\cdot[A^{2},A^{2},0]$ and $[A^{2},A^{2},A]\cdot[A^{2},A^{2},A],$ respectively. We have the following theorems.

Theorem 1.3**.**

If $A\subset\mathbb{F}_{p}$ with $|A|\leq p^{\frac{1}{2}}$, then we have

[TABLE]

Theorem 1.4**.**

Let $A\subset\mathbb{F}_{p}$ with $|A|\leq p^{\frac{9}{16}}.$ Then we have

[TABLE]

The rest of this paper is organized to provide the complete proofs of our main theorems. More precisely, in Section $2$ we give the proof of Theorem 1.1, and in Section $3$ we complete proofs of Theorems 1.3 and 1.4.

2 Proof of Theorem 1.1

In this section, without loss of generality, we assume that $0\not\in A$. To prove Theorem 1.1, we need the following lemmas.

Lemma 2.1** ([16], Corollary $3.1$).**

Let $X,A\subset\mathbb{F}_{p}$ with $|X|\geq|A|$. Then we have

[TABLE]

Lemma 2.2**.**

Let $A\subset\mathbb{F}_{p}$ with $|A|\leq cp^{2/3}$ for a sufficiently small $c>0.$ Then the number of tuples $(a_{1},a_{2},a_{3},a_{4},a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime},a_{4}^{\prime})\in A^{8}$ satisfying

[TABLE]

is $\ll|A|^{13/2}$

Proof.

For $\lambda,\beta\in\mathbb{F}_{p}\setminus\{0\}$, one can follow the proof of [9, Theorem $3$] to prove that the number of tuples $(a_{1},a_{2},a_{3},a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})\in A^{6}$ such that

[TABLE]

is $\ll|A|^{9/2}$. Thus we see that for each fixed pair $(a_{4},a_{4}^{\prime})\in A^{2}$ the number of tuples $(a_{1},a_{2},a_{3},a_{4},a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime},a_{4}^{\prime})\in A^{8}$ satisfying

[TABLE]

is $\ll|A|^{9/2}$. Taking the sum over all pairs $(a_{4},a_{4}^{\prime})\in A^{2}$, the lemma follows.

∎

Lemma 2.3** ([11], Theorem 4).**

Let $A,B\subset\mathbb{F}_{p}$ with $|A|\leq|B|,$ and let $L$ be a finite set of lines in $\mathbb{F}_{p}^{2}.$ Suppose that $|A||B|^{2}\leq|L|^{3}$ and $|A||L|\ll p^{2}.$ Then the number of incidences between $A\times B$ and lines in $L$, denoted by $I(A\times B,L)$, satisfies

[TABLE]

The following is an improvement of Lemma $23$ in [15].

Lemma 2.4**.**

Let $A,B\subset\mathbb{F}_{p}$. Then if $|A|=|B|$, and $|A|^{2}|AB|\ll p^{2}$, we have

[TABLE]

for any $x\neq 0$.

Proof.

It is clear that

[TABLE]

The number of such tuples $(u,u_{*},a,b)$ is bounded by the number of incidences between points in $A^{-1}\times AB$ and a set $L$ of lines of the form $b^{-1}Y-u_{*}X=x$ with $b\in B$ and $u_{*}\in AB$. Notice that $|A|=|A^{-1}|$ and $|L|=|B||AB|.$ Thus if $|A|=|B|$ and $|A|^{2}|AB|\ll p^{2}$, Lemma 2.3 implies that

[TABLE]

which completes the proof of the theorem. ∎

Lemma 2.5** ([8], Theorem 2).**

If $A\subset\mathbb{F}_{p}$ with $|A|\leq p^{9/16}$, then we have

[TABLE]

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1.

Without loss of generality, we may assume that $0\notin A.$ Let $M_{1}$ and $M_{2}$ be matrices in $R(A)$ presented as follows:

[TABLE]

Suppose that

[TABLE]

where $t\neq 0$ and $\alpha,\beta\in\mathbb{F}_{p}.$ Then we have the following system

[TABLE]

Let us identify the matrix $M_{1}\cdot M_{2}$ with $(t,\alpha,\beta)\in\mathbb{F}_{p}^{*}\times\mathbb{F}_{p}^{2}.$ Notice that $R(A)\cdot R(A)$ contains each $(t,\alpha,\beta)\in\mathbb{F}_{p}^{*}\times\mathbb{F}_{p}^{2}$ satisfying the system (1) for some $(a_{11},a_{12},a_{21},b_{11},b_{12},b_{21})$ in $A^{6}.$ Therefore, we aim to estimate a lower bound of the number of $(t,\alpha,\beta)\in\mathbb{F}_{p}^{*}\times\mathbb{F}_{p}^{2}$ such that the system (1) holds for some $(a_{11},a_{12},a_{21},b_{11},b_{12},b_{21})\in A^{6}.$ To this end, let $\epsilon>0$ be a parameter chosen later. We now consider two following cases:

If $|AA|\geq|A|^{1+\epsilon}$, then it follows from Lemma 2.1 that

[TABLE]

where we assume that

[TABLE]

From the system (1) and the above fact, we obtain that if $|A|\leq p^{\frac{2}{3+\epsilon}}$ and $|AA|\geq|A|^{1+\epsilon}$, then

[TABLE]

where the first $\gg$ follows, because in the system (1), for each non-zero $t\in AA+AA$, if we fix a quadruple $(a_{11},b_{11},a_{12},b_{21})\in A^{4}$ with $a_{11}b_{11}+a_{12}b_{21}=t$, then $\alpha,\beta$ are determined in terms of $b_{12}\in A$ and $a_{21}\in A$, respectively. 2. 2.

If $|AA|\leq|A|^{1+\epsilon}$, then we consider as follows. For $t,\alpha,\beta\in\mathbb{F}_{p}$, let $\nu(t,\alpha,\beta)$ be the number of solutions $(a_{11},a_{12},a_{21},b_{11},b_{12},b_{21})$ of the system (1). For the case $t=0$, we have

[TABLE]

Indeed, for each choice of $(b_{11},a_{12},b_{21})\in A^{3}$, $a_{11}$ is determined uniquely, and $\alpha,\beta$ are determined. In addition, $a_{21}$ and $b_{12}$ can be taken as arbitrary elements of $A.$

By the Cauchy-Schwarz inequality, we have

[TABLE]

This implies that

[TABLE]

where $\displaystyle T:=\sum_{t\neq 0,\alpha,\beta}\nu^{2}(t,\alpha,\beta)$.

In the next step, we are going to show that

[TABLE]

To see this, observe by definition of $\nu(t,\alpha,\beta)$ that for each $(t,\alpha,\beta)\in\mathbb{F}_{p}^{*}\times\mathbb{F}_{p}^{2},$ the value $\nu^{2}(t,\alpha,\beta)$ is the number of 12-tuples $(a_{11},a_{12},a_{21},b_{11},b_{12},b_{21},a^{\prime}_{11},a^{\prime}_{12},a^{\prime}_{21},b^{\prime}_{11},b^{\prime}_{12},b^{\prime}_{21})\in A^{12}$ satisfying the following:

[TABLE]

Thus the value of $\displaystyle T=\sum_{t\neq 0,\alpha,\beta}\nu^{2}(t,\alpha,\beta)$ can be written by $\displaystyle\sum_{t\neq 0}\Omega(t)$ where $\Omega(t)$ denotes the number of 12-tuples $(a_{11},a_{12},a_{21},b_{11},b_{12},b_{21},a^{\prime}_{11},a^{\prime}_{12},a^{\prime}_{21},b^{\prime}_{11},b^{\prime}_{12},b^{\prime}_{21})\in A^{12}$ satisfying the following:

[TABLE]

Now notice that Lemma 2.2 implies that if $|A|\ll p^{2/3},$ then there are at most $|A|^{13/2}$

$8$-tuples $(a_{11},b_{11},a_{12},b_{21},a^{\prime}_{11},b^{\prime}_{11},a^{\prime}_{12},b^{\prime}_{21})$ in $A^{8}$ satisfying the equations (4) for some $t\neq 0.$ One can also check that among these tuples, there are at most $|A|^{6}$ $(\leq\frac{|A|^{13/2}}{2})$ tuples with $a^{\prime}_{12}b^{\prime-1}_{11}-a_{12}b^{-1}_{11}=0$. Hence, without loss of generality, we may assume that all tuples satisfy $a^{\prime}_{12}b^{\prime-1}_{11}-a_{12}b^{-1}_{11}\neq 0$.

For such a fixed $8$-tuple $(a_{11},b_{11},a_{12},b_{21},a^{\prime}_{11},b^{\prime}_{11},a^{\prime}_{12},b^{\prime}_{21})\in A^{8}$, we now deal with the equation (5) which can be rewritten by

[TABLE]

Set $Q=\frac{t}{b_{11}}\cdot A$,  $Q^{\prime}=\frac{t}{b^{\prime}_{11}}\cdot A$, and $x=a_{12}b^{-1}_{11}-a^{\prime}_{12}b^{\prime-1}_{11}\neq 0$. Then the number of solutions $(b_{12},b_{12}^{\prime})\in A^{2}$ of (7) is the size of $Q\cap(Q^{\prime}-x)$. It is clear that $|Q|=|Q^{\prime}|=|A|,$ because $t\neq 0$ and we have assumed that $0\notin A$ so that $t/b_{11},t/b^{\prime}_{11}\neq 0.$ We also see that

[TABLE]

where we used the assumption that $|AA|\leq|A|^{1+\epsilon}.$ Applying Lemma 2.4, we obtain that if $|A|\ll p^{2/(3+\epsilon)},$ then

[TABLE]

The same argument works identically for the equation (6) which can be restated by

[TABLE]

In short, we have proved that if $|A|\ll p^{2/(3+\epsilon)}$ and $|AA|\leq|A|^{1+\epsilon}$, then

[TABLE]

Therefore, combining (3) with this estimate yields that if $|A|\ll p^{2/(3+\epsilon)}$ and $|AA|\leq|A|^{1+\epsilon}$, then

[TABLE]

Finally, if we choose $\epsilon=1/6$, then it follows from (2) and (9) that if $|A|\ll p^{12/19}$, then

[TABLE]

which completes the proof of Theorem 1.1. ∎

In the case of arbitrary finite fields, we have the following result.

Theorem 2.6**.**

Let $q=p^{n}$ and let $A$ be a subset of $\mathbb{F}^{*}_{q}$. If $|A\cap\lambda F|\leq|F|^{1/2}$ for any proper subfield $F$ of $\mathbb{F}_{q}$ and any $\lambda\in\mathbb{F}_{q}$, then we have

[TABLE]

To prove Theorem 2.6 we make use of the following results.

Theorem 2.7** ([7]).**

With the assumptions of Theoem 2.6, we have

[TABLE]

Theorem 2.8**.**

For $A,B\subset\mathbb{F}_{q}$, suppose that $|A\cap\lambda F|,|B\cap\lambda F|\leq|F|^{1/2}$ for any subfield $F$ of $\mathbb{F}_{q}$ and any $\lambda\in\mathbb{F}_{q}$. Then we have

[TABLE]

Corollary 2.9**.**

For $A\subset\mathbb{F}_{q}$, suppose that $|A\cap\lambda F|\leq|F|^{1/2}$ for any subfield $F$ of $\mathbb{F}_{q}$ and any $\lambda\in\mathbb{F}_{q}$. Then we have

[TABLE]

Proof.

Given a nonzero $x\in A$, we have $|AA+AA|=|\frac{A}{x}\frac{A}{x}+\frac{A}{x}\frac{A}{x}|\geq|\frac{A}{x}\frac{A}{x}+\frac{A}{x}|\gg|A|^{6/5}$ by Theorem 2.8.

∎

we are now ready to prove Theorem 2.6.

Proof of Theorem 2.6.

Recall from (2) that

[TABLE]

Thus the Theorem follows directly from Corollary 2.9. ∎

In the rest of this section, we present the proof of Theorem 2.8, for which the authors communicated with Oliver Roche-Newton.

2.1 Proof of Theorem 2.8

To prove Theorem 2.8, we make use of the following lemmas.

The first lemma is the Plünnecke-Ruzsa inequality.

Lemma 2.10** ([10], Theorems 1.6.1, 1.81).**

Let $X,B_{1},\ldots,B_{k}$ be subsets of $\mathbb{F}_{q}$. Then we have

[TABLE]

and

[TABLE]

One can modify the proof of Corollary $1.5$ in [6] due to Katz and Shen to obtain the following.

Lemma 2.11**.**

Let $X,B_{1},\ldots,B_{k}$ be subsets in $\mathbb{F}_{q}$. Then, for any $0<\epsilon<1$, there exists a subset $X^{\prime}\subset X$ such that $|X^{\prime}|\geq(1-\epsilon)|X|$ and

[TABLE]

for some positive constant $c=c(\epsilon).$

We also have the following lemma from [7].

Lemma 2.12**.**

Let $B$ be a subset of $\mathbb{F}_{q}$ with at least two elements, and define $\mathbb{F}_{B}$ as the subfield generated by $B$. Then there exists a polynomial $P(x_{1},\ldots,x_{n})$ of several variables with coefficients belonging to the prime field $\mathbb{F}_{p}$ such that

[TABLE]

We are now ready to prove Theorem 2.8.

Proof of Theorem 2.8.

WLOG, we may assume $1\in A$ by considering $\frac{1}{x}A$ for some $x\in A$. We first define the ratio set:

[TABLE]

We now consider the following cases:

Case 1: $1+R(A,B)\not\subset R(A,B)$.

In this case, there exist $a_{1},a_{2}\in A$ and $b_{1}\neq b_{2}\in B$ such that

[TABLE]

First, we apply Lemma 2.11 so that there exists a subset $A^{\prime}\subset A$ such that $|A^{\prime}|\gg|A|$ and

[TABLE]

On the other hand, we have

[TABLE]

Since $r\not\in R(A,B)$, the equation

[TABLE]

has no non-trivial solutions, i.e. solutions $(a_{1},b_{1},a_{2},b_{2})$ with $b_{1}\neq b_{2}$. This implies that

[TABLE]

We now give an upper bound for $(b_{1}-b_{2})A+(a_{1}-a_{2})B=b_{1}A+a_{1}B-b_{2}A-a_{2}B$ which will be used in the rest of the proof.

Lemma 2.11 tells us that there exists a subset $X\subset A$ such that $|X|\gg|A|$ and

[TABLE]

and there exists a subset $X^{\prime}\subset X$ with $|X^{\prime}|\gg|X|$ such that

[TABLE]

Applying Lemma 2.10, we have

[TABLE]

Putting (10-13) together, we obtain

[TABLE]

and we are done.

Case 2: $B\cdot R(A,B)\not\subset R(A,B)$. Similarly, in this case, there exist $a_{1},a_{2}\in A$ and $b,b_{1},b_{2}\in B$ such that

[TABLE]

Since $0\in R(A,B)$, we have $b\neq 0$, and $a_{1}\neq a_{2}$. This gives us that $r^{-1}$ exists.

Using the same argument as above, we have

[TABLE]

Thus we obtain

[TABLE]

and we done.

Case 3: $B^{-1}\cdot R(A,B)\not\subset R(A,B)$.

As above, in this case, there exist $a_{1},a_{2}\in A$ and $b\neq 0,b_{1}\neq b_{2}\in B$ such that

[TABLE]

Since $0\in R(A,B)$, we have $a_{1}\neq a_{2}$. This gives us that $r^{-1}$ exists. The rest is the same as the Case $2$.

Case 4: We consider the case when

[TABLE]

Now we are going to show that for any polynomial $P(x_{1},\ldots,x_{n})$ in $n$ variables, for some positive integer $n$, and coefficients belonging $\mathbb{F}_{p}$ such that

[TABLE]

Indeed, it is enough to show that

[TABLE]

for any integer $d\geq 1$, and $B^{d}=B\cdots B$ ($d$ times).

It follows from the assumption that the first condition $1+R(A,B)\subset R(A,B)$ is satisfied. For the second condition, it is sufficient to prove it for $d=2$, since we can use inductive arguments.

Let $b,b^{\prime}$ be arbitrary elements in $B$. We now show that

[TABLE]

If either $b=0$ or $b^{\prime}=0$, then we are done. Thus we may assume that $b\neq 0$ and $b^{\prime}\neq 0$.

First we have

[TABLE]

and

[TABLE]

In other words, for any polynomial $P(x_{1},x_{2},\ldots,x_{n})\in\mathbb{F}_{p}[x_{1},\ldots,x_{n}]$ we have

[TABLE]

Furthermore, Lemma 2.12 tells us that there exists a polynomial $P$ such that $P(B,\ldots,B)=\mathbb{F}_{B}$.

This implies that

[TABLE]

It follows from our assumption of the theorem that

[TABLE]

Hence, $|R(A,B)|\geq|\mathbb{F}_{B}|\geq|B|^{2}$.

Next we shall show that there exists $r\in R(A,B)$ such that either

[TABLE]

or

[TABLE]

Indeed, let $E^{+}(X,Y)$ be the number of tuples $(x_{1},x_{2},y_{1},y_{2})\in X^{2}\times Y^{2}$ such that

[TABLE]

We have the sum $\sum_{r\in R(A,B)}E^{+}(A,rB)$ is the number of tuples $(a_{1},a_{2},b_{1},b_{2})\in A^{2}\times B^{2}$ such that

[TABLE]

with $a_{1},a_{2}\in A$, $b_{1},b_{2}\in B$ and $r\in R(A,B)$. It is easy to see that there are at most $|R(A,B)||A||B|$ tuples with $a_{1}=a_{2},b_{1}=b_{2}$, and at most $|A|^{2}|B|^{2}$ tuples with $b_{1}\neq b_{2}$. Therefore, we get

[TABLE]

By the pigeon-hole principle, there exists $r:=\frac{a_{1}-a_{2}}{b_{1}-b_{2}}\in R(A,B)$ such that

[TABLE]

So, either

[TABLE]

or

[TABLE]

We now fall into two small cases:

If $|A+rB|\gg|A||B|$, then, applying Lemma 2.10, we have

[TABLE]

which gives us

[TABLE]

• 2.

If $|A+rB|\gg|B|^{2}$, then we have

[TABLE]

which gives us

[TABLE]

This completes the proof of the theorem.

∎

3 Proofs of Theorems 1.3 and 1.4

In the proof of Theorem 1.3, we make use of the following version of Balog-Szemerédi-Gowers theorem due to Schoen [12].

Theorem 3.1** ([12], Theorem 1.1).**

Let $G$ be an abelian group. Suppose that $A$ is a subset of $G$, and $E^{+}(A)$ denotes the additive energy which is the number of solutions $(a,b,c,d)\in A^{4}$ to the equation $a+b=c+d.$ If $E^{+}(A)$ is equal to $k|A|^{3}$, then there exists $A^{\prime}\subset A$ with $|A^{\prime}|\gg k|A|$ such that

[TABLE]

We also will need the following results.

Theorem 3.2**.**

For $A,B,C,D\subset\mathbb{F}_{p}$, let $Q(A,B,C,D)$ be the number of $8$-tuples

[TABLE]

such that

[TABLE]

We have

[TABLE]

where

[TABLE]

To prove this theorem, we need the following version of point-plane incidence bound due to Rudnev in [8].

Theorem 3.3** ([8]).**

Let $P$ be a set of points in $\mathbb{F}_{p}^{3}$ and let $\Pi$ be a set of planes in $\mathbb{F}_{p}^{3}$. Suppose that $|P|\leq|\Pi|$, and there are at most $k$ collinear points in $P$ for some $k$, then the number of incidences between $P$ and $\Pi$ is bounded by

[TABLE]

we are now ready to prove Theorem 3.2. We will follow the ideas of the proof of [13, Theorem $32$].

Proof of Theorem 3.2.

We have

[TABLE]

where $r_{CD}(\lambda)$ is the number of pairs $(c,d)\in C\times D$ such that $cd=\lambda$, $r_{AB}(\mu)$ is the number of pairs $(a,b)\in A\times B$ such that $ab=\mu$, and $n(\lambda,\mu)=\sum_{x}r_{AB+\lambda}(x)r_{CD+\mu}(x)$. If we split the sum $Q(A,B,C,D)$ into intervals, we get

[TABLE]

where $L_{1}\leq\log(|C||D|),L_{2}\leq\log(|A||B|)$, $r_{AB}^{(i)}(\mu)$ is the restriction of the function $r_{AB}(x)$ on the set $P_{i}:=\{\mu\colon\Delta_{i}\leq r_{AB}(\mu)<2\Delta_{i}\}$, and $r_{CD}^{(i)}(\lambda)$ is the restriction of the function $r_{CD}(x)$ on the set $P_{i}:=\{\lambda\colon\Delta_{i}\leq r_{CD}(\lambda)<2\Delta_{i}\}$. Applying the pigeon-hole principle two times, there exist sets $P_{i}$ and $P_{j}$ such that

[TABLE]

where $P_{i}(x)$ is the indicator function of the set $P_{i}$. For the simplicity, we suppose that $i=1$ and $j=2$.

One can check that the sum $\sum_{\lambda,\mu}n(\lambda,\mu)P_{1}(\lambda)P_{2}(\mu)$ is the number of incidences between points $(a,d,\lambda)\in A\times D\times P_{1}\subset\mathbb{F}_{p}^{3}$ and planes in $\mathbb{F}_{p}^{3}$ defined by

[TABLE]

where $b\in B,c\in C,\mu\in P_{2}$.

With the way we define the plane set, it follows from [17] that we can apply Theorem 3.3 with $k=\max\{|A|,|D|\}$. Thus we obtain

[TABLE]

It is clear in our argument that we can switch the point set and the plane set, we also can do the same thing for $P_{1}$ and $P_{2}$ in the definition of the point set and the plane set. So, without loss of generality, we can assume that $|P_{1}|\leq|P_{2}|$, $|A||D|\leq|B||C|$. We now consider the following cases:

• •

If the second term dominates, then we have

[TABLE]

since

[TABLE]

• •

If the first term dominates, then we have

[TABLE]

since

[TABLE]

• •

If the last term dominates, then we study the following:

Suppose $|A|\leq|D|$. If $|D|\leq|P_{2}|$, then it is easy to check that the second term in (17) will be bigger than the last term. Thus, we can suppose that $|D|\geq|P_{2}|$. Since $|P_{1}|\leq|P_{2}|$, we have

[TABLE]

On the other hand, it is clear that $\Delta_{1}\Delta_{2}\leq|B||C|$. This means

[TABLE] 2. 2.

Suppose $|A|\geq|D|$. By repeating the same argument, we obtain

[TABLE]

This completes the proof of the theorem. ∎

Proof of Theorem 1.3.

Let $N$ be the number of tuples

[TABLE]

such that $[\mathbf{x},\mathbf{y},0]\cdot[\mathbf{z},\mathbf{t},0]=[\mathbf{x^{\prime}},\mathbf{y^{\prime}},0]\cdot[\mathbf{z^{\prime}},\mathbf{t^{\prime}},0]$. This can be expressed as follows:

[TABLE]

Thus by the Cauchy-Schwarz inequality, we have

[TABLE]

From (18), observe that $N$ is the number of tuples

[TABLE]

satisfying the following system:

[TABLE]

We now insert new variables $s_{1},s_{2},s_{3},s_{4}\in A+A$ in (20) and (21) as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Set $A_{s_{i}}=A\cap(s_{i}-A)$, then we have $x_{1},x_{1}^{\prime}\in A_{s_{1}}$, $x_{2},x_{2}^{\prime}\in A_{s_{1}}$, $y_{1},y_{1}^{\prime}\in A_{s_{3}}$, $y_{2},y_{2}^{\prime}\in A_{s_{4}}$.

In this setting, we have

[TABLE]

Applying Theorem 3.2, and assuming the second and third terms are larger than the first term, we have

[TABLE]

where we have used the fact that

[TABLE]

Moreover, using the fact $E^{\times}(A,B)\leq|A|^{2}|B|$, we have

[TABLE]

In other words, we have proved that

[TABLE]

If $N\lesssim|A|^{10}$, then the theorem follows from (19). Therefore, we can assume that

[TABLE]

Let $\epsilon$ be a parameter chosen later. We now consider two cases:

Suppose that $E^{+}(A)<|A|^{3-\epsilon}.$ Then we have

[TABLE]

From (19), this implies that

[TABLE] 2. 2.

Suppose that $E^{+}(A)\geq|A|^{3-\epsilon}.$ Then we can write $E^{+}(A)=|A|^{3-\epsilon^{\prime}}$ for some $\epsilon^{\prime}<\epsilon<1.$ Notice that Theorem 3.1 implies that there exists a subset $A^{\prime}\subset A$ such that $|A^{\prime}|\gg|A|^{1-\epsilon}$ and

[TABLE]

Since $|A|\leq p^{9/16}$ by our assumption, using Lemma 2.5 with the above inequality gives

[TABLE]

Moreover, one can easily check that

[TABLE]

Therefore,

[TABLE]

Moreover, Lemma 2.1 gives us

[TABLE]

where we also utilized the inequality (26) and the fact that $|A^{\prime}|\gg|A|^{1-\epsilon}.$

Thus we obtain that if $|A|\leq p^{9/16},$ then

[TABLE]

provided that $|A|^{1+(1-\epsilon)(\frac{7}{9}-\frac{4\epsilon}{1-\epsilon})}\leq p.$ It is clear that with $\epsilon=5/131$, this estimate is satisfied since $|A|\leq p^{1/2}$. We also obtain

[TABLE]

Finally, suppose the first term is the largest term. Then we have $N\leq\frac{E^{+}(A,A)^{4}}{p}$ which is smaller than $\frac{|A|^{12}}{p}$. Thus (19) shows that

[TABLE]

This is more than what we want and completes the proof of the theorem. ∎

Over finite arbitrary fields $\mathbb{F}_{q}$, we have the following result.

Theorem 3.4**.**

Let $q=p^{n}$ and let $A$ be a subset of $\mathbb{F}^{*}_{q}$. If $|A\cap\lambda F|\leq|F|^{1/2}$ for any proper subfield $F$ of $\mathbb{F}_{q}$ and any $\lambda\in\mathbb{F}_{q}$, then we have

[TABLE]

and

[TABLE]

Proof.

We first observe that

[TABLE]

and

[TABLE]

It follows from Theorem 2.7 and 2.8 that

[TABLE]

Therefore, we obtain

[TABLE]

and

[TABLE]

which completes the proof of the theorem. ∎

Proof of Theorem 1.4.

One can observe that

[TABLE]

We now prove that if $|A|\leq p^{9/16},$ then

[TABLE]

Indeed, we first prove that if $|A|\leq p^{9/16},$

[TABLE]

To prove this inequality, we consider the following cases:

If $|A+A|\geq|A|^{1+\epsilon}$, then it follows from Lemma 2.1 that

[TABLE]

whenever

[TABLE] 2. 2.

If $|A+A|\leq|A|^{1+\epsilon}$, then Lemma 2.5 gives us that $|AA|\gtrsim|A|^{\frac{14}{9}-2\epsilon}$ under the condition $|A|\leq p^{9/16}$. Hence, if $|A|\leq p^{9/16},$ then

[TABLE]

Choosing $\epsilon=1/45$, we see from (29) and (30) that if $|A|\leq p^{45/68}$ and $|A|\leq p^{9/16}$, then

[TABLE]

Since $p^{45/68}\geq p^{9/16},$ we establish the inequality (28).

By Lemma 2.1 and the inequality (28), we see that if $|A|\leq p^{9/16}$, then

[TABLE]

Finally, combining (27) and this estimate, we conclude that if $|A|\leq p^{9/16}$, then

[TABLE]

which completes the proof of Theorem 1.4 . ∎

Acknowledgments

The authors would like to deeply thank Dr. Oliver Roche-Newton and Dr. Ilya Shkredov for many helpful discussions that make nice improvement for our Theorem 1.3.

D. Koh was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(NRF-2015R1A1A1A05001374). T. Pham was supported by Swiss National Science Foundation grant P2ELP2175050. C-Y Shen was supported in part by MOST, through grant 104-2628-M-002-015 -MY4.