Sum-Product Type Estimates over Finite Fields

TL;DR

This paper establishes sum-product estimates over finite fields using Fourier analysis and third moment methods, showing that large subsets have product-plus-sum sets nearly as large as the entire field.

Contribution

It introduces a novel combination of Fourier analytic tools and third moment techniques to derive new sum-product estimates over finite fields.

Findings

For subsets A of finite fields, |AA+A| and |A(A+A)| are large, at least proportional to min{q, |A|^2/q^{1/2}}.

If |A| ≥ q^{3/4}, then |AA+A| and |A(A+A)| are at least proportional to q.

The methods improve understanding of sum-product phenomena in finite fields.

Abstract

Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of $\mathbb{F}_q$. In particular, we prove that if $A\subset \mathbb{F}_q$, then $$|AA+A|,|A(A+A)|\gg\min\left\{q, \frac{|A|^2}{q^{\frac{1}{2}}} \right\},$$ so that if $A\ge q^{\frac{3}{4}}$, then $|AA+A|,|A(A+A)|\gg q$.

Equations

$$|AA+A|,|A(A+A)|\gg \min \left\{q,\frac{|A{|}^2}{q^{\frac{1}{2}}}\right\},$$

$$A+A=\{a+b:a,b\in A\}\text{and}AA=\{a.b:a,b\in A\},$$

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

$$|A+A|+|AA|\gg |A{|}^{\frac{4}{3}+\frac{5}{5277}}$$

$$\hat{f}(m)=q^{-1}\sum _{x\in {\mathbb{F}}_q}\chi (-xm)f(x)$$

$$\sum_{x\in\mathbb{F}_{q}}\chi(xs)=\left\{\begin{array}[]{rl}q,&\text{if }s=0\\ 0,&otherwise\end{array}\right.$$

$$\sum _{m\in {\mathbb{F}}_q}|\hat{f}(m){|}^2=q^{-1}\sum _{x\in {\mathbb{F}}_q}|f(x){|}^2.$$

$$|BA+C|,|B(A+C)|\gg \min \left\{q,\frac{|B{|}^{\frac{1}{2}}|C{|}^{\frac{1}{2}}|A|}{q^{\frac{1}{2}}}\right\}$$

$$|AA+A|,|A(A+A)|\gg \min \left\{q,\frac{|A{|}^2}{q^{\frac{1}{2}}}\right\}.$$

$$L=L_P=\{l_{m,b}:(m,b)\in P\}$$

$$L(A)=L_P(A)=\{l_{m,b}(a)=ma+b:(m,b)\in P,a\in A\}.$$

$$E_3(L,A)=|\{m_1{a}_1+b_1=m_2{a}_2+b_2=m_3{a}_3+b_3\}|$$

$$\frac{|L{|}^3|A{|}^3}{|L(A){|}^2}\le E_3(L,A).$$

$$|L||A|=\sum r(x)$$

$$\frac{|L{|}^3|A{|}^3}{|L(A){|}^2}$$

$$\sum _{z\in F}{f}^n(z)\le |F|{\left(\frac{||f|{|}_1}{|F|}\right)}^n+\frac{n(n-1)}{2}||f|{|}_{\infty }^{n-2}\sum _{z\in F}{\left(f(z)-\frac{||f|{|}_1}{|F|}\right)}^2$$

$$L(A)\gg \min \left\{q,\frac{|L{|}^{\frac{1}{2}}|A|}{q^{\frac{1}{2}}}\right\}.$$

$$f(z):=r_{L(A)}(z)=|\{z=a_1{a}_2+a_3:(a_1,a_3)\in P,a_2\in A\}|$$

$$E_3(L,A)=\sum _{z}f(z{)}^3\le \frac{||f|{|}_1^3}{q^2}+3||f|{|}_{\infty }\sum _{z\in F}{\left(f(z)-\frac{||f|{|}_1}{q}\right)}^2.$$

$$E_3(L,A)$$

$$f(z)$$

$$\hat{f}(\xi )$$

$$|\hat{f}(\xi )|\le \sum _{a_1\in B}|\sum _{a_2\in A}\chi (-a_1{a}_2\xi )\hat{C}(\xi )|$$

$$|\hat{f}(\xi ){|}^2$$

$$\sum _{\xi \ne 0}|\hat{f}(\xi ){|}^2$$

$$\frac{|L{|}^3|A{|}^3}{|L(A){|}^2}\le E_3(L,A)\le \frac{|L{|}^3|A{|}^3}{q^2}+3|L{|}^2|A|q$$

$$L(A)\gg \min \left\{q,\frac{|L{|}^{\frac{1}{2}}|A|}{q^{\frac{1}{2}}}\right\}.$$

$$|BA+C|\gg \min \left\{q,\frac{|B{|}^{\frac{1}{2}}|C{|}^{\frac{1}{2}}|A|}{q^{\frac{1}{2}}}\right\}$$

Full text

In particular, we prove that if $A\subset\mathbb{F}_{q}$, then

[TABLE]

so that if $A\geq q^{\frac{3}{4}}$, then $|AA+A|,|A(A+A)|\gg q$.

1 Introduction

Let $R$ be a ring. For a finite subset $A$ of $R$ we define the sum set and the product set of $A$ by

[TABLE]

respectively. It is expected that, if $A$ is not subring of $R$, then either $|A+A|$ or $|A.A|$ is large compared to $|A|$.

In [4], Erdős and Szemeredi proved that that there exists an absolute constant $\epsilon>0$ such that

[TABLE]

holds for any finite subset $A$ of $\mathbb{Z}$. They also conjectured that this bound should hold for any $\epsilon<1$. The best known bound in this direction is due to Shakan [11, Theorem 1.2] which states that if $A$ is a finite subset of $\mathbb{R}$, then

[TABLE]

The sum-product problem in the finite field context has been studied by various authors. In this setting, one generally works either on the small sets in terms of the characteristic $p$ of $\mathbb{F}_{q}$ or for sufficiently large subsets of $\mathbb{F}_{q}$ to guarantee that the set itself is in fact not a proper subfield of $\mathbb{F}_{q}$. We refer the reader [1, 3, 5, 6, 7, 8, 9] and the references therein for an extensive exploration of the problem in this context.

In the present paper, we turn our attention to sum-product type estimates for the sets of the form $BA+C=\{ba+c:a\in A,b\in B,c\in C\}$ and $B(A+C)=\{b(a+c):a\in A,b\in B,c\in C\}$ where $A,B$, and $C$ are subsets of $\mathbb{F}_{q}$. To estimate a lower bound for these sets, we first consider an additive energy which we relate with a third moment method. Then we employ a lemma from [2] and prove the main result in the paper using the tools in Fourier analysis.

1.1 Preliminaries

Let $f:\mathbb{F}_{q}\to\mathbb{C}.$ The Fourier transform of $f$ is defined as

[TABLE]

where $\chi(z)=e^{\frac{2\pi iz}{q}}$. We will use the orthogonality relation

[TABLE]

and Plancherel identity

[TABLE]

The main result of the paper is the following theorem.

Theorem 1.1**.**

If $A,B,C\subset\mathbb{F}_{q}$, then

[TABLE]

In particular, taking $A=B=C,$ we have

[TABLE]

so that if $A\geq q^{\frac{3}{4}}$, then $|AA+A|,|A(A+A)|\gg q$.

1.2 Proof of Theorem 1.1

Let $A\subset\mathbb{F}_{q}$ and $P$ be a set of points in $\mathbb{F}_{q}^{2}\setminus\{(0,0)\}.$ Define the set of lines pinned at $P$ as

[TABLE]

and also the image set of lines in $L$ as

[TABLE]

Similar to energy notion given in [1], define

[TABLE]

where $(m_{1},b_{1}),(m_{2},b_{2}),(m_{3},b_{3})\in P,a_{i}\in A.$

Lemma 1.2**.**

With the notation above we have

[TABLE]

Proof.

Let $r(x)=r_{L(A)}(x)=|\{((m,b),a)\in P\times A:x=ma+b\}|$. Then, by Hölder inequality,

[TABLE]

∎

We need the following lemma from [2].

Lemma 1.3**.**

[2, Lemma 2.1]**

$F$* a finite space, $f:F\to R$.*

[TABLE]

where $||f||_{1}=\sum_{z\in F}|f(z)|$, $||f||_{\infty}=\max_{z\in F}f(z)$.

Theorem 1.4**.**

Let $L=L_{P}$ where $P\cong B\times C$.

[TABLE]

Proof.

Now let

[TABLE]

Then by taking $n=3$, $F=\mathbb{F}_{q}$ in Lemma 1.3, we have

[TABLE]

Note that $||f||_{1}=\sum f(z)=|L||A|$, $||f||_{\infty}=sup_{z}f(z)\leq|L|$, since when we fix $(a_{1},a_{3})$ in $f(z)$ then $a_{2}$ is uniquely determined.

Therefore,

[TABLE]

where we used the Plancherel in the last equality.

We can write

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It follows that

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Therefore,

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By the Cauchy-Schwarz inequality, for $\xi\neq 0$,

[TABLE]

It follows that

[TABLE]

Plugging the last value in (1) and using Lemma 1.2 we have

[TABLE]

Therefore

[TABLE]

∎

Proof of Theorem 1.1.

Note that the set $BA+C=L_{P}(A)$ where $P=B\times C$. Hence, taking $|L|=|B||C|$ in Theorem 1.4, it follows that

[TABLE]

Note that $B(A+C)=L_{P}(A)$ where $P\cong B\times C$, so the same argument applies. ∎

Acknowledgements

The author would like to thank Simon Macourt, Oliver Roche-Newton, Alex Iosevich, Jonathan Pakianathan and Ilya Shkredov for their valuable comments. The author is supported by TÜBİTAK-BİDEB 2218 Postdoctoral Research Fellowship.