Conditional expanding bounds for two-variable functions over arbitrary fields

TL;DR

This paper improves bounds on how two-variable functions expand over arbitrary fields by applying Rudnev's point-plane incidence bound, refining previous results in the area.

Contribution

It introduces an improved bound on conditional expanding bounds for two-variable functions over arbitrary fields using Rudnev's incidence theorem.

Findings

Enhanced bounds on expanding functions over arbitrary fields

Application of Rudnev's point-plane incidence bound to this problem

Refinement of previous results by Hegyvári and Hennecart

Abstract

In this short note, we use Rudnev's point-plane incidence bound to improve some results on conditional expanding bounds for two-variable functions over arbitrary fields due to Hegyv\'ari and Hennecart.

Equations

$$A+A=\{a+a^{\prime }:a,a^{\prime }\in A\},A\cdot A=\{a\cdot a^{\prime }:a,a^{\prime }\in A\}.$$

$$\max \left\{|A+A|,|A\cdot A|\right\}\gg |A{|}^{1+ϵ},$$

$$\max \left\{|A\pm A|,|A\cdot A|\right\}\gg |A{|}^{6/5},\max \left\{|A\pm A|,|A:A|\right\}\gg |A{|}^{6/5}.$$

$$\mu (g)=\underset{t\in {\mathbb{F}}^{\ast }}{\max }\left|\left\{x\in G:g(x)=t\right\}\right|.$$

$$\max \left\{|f(A,B)|,|g(A,B)|\right\}\gg |A{|}^{1+\Delta (\alpha )},$$

$$\left|f(A,B)\right|\left|B\cdot C\right|\gg \min \left\{\frac{|A||B{|}^2|C|}{p{m}^2},\frac{p|B|}{m}\right\}.$$

$$|f(A,B)||B+C|\gg \min \left\{\frac{|A||B{|}^2|C|}{p{m}^2},\frac{p|B|}{m}\right\}$$

$$|f(A,A)||A\cdot A|,|f(A,A)||A+A|\gg p|A|.$$

$$|f(A,A)||A\cdot A|,|f(A,A)||A+A|\gg |A{|}^4/p.$$

$$\max \left\{|f(A,B)|,|B\cdot C|\right\}\gg \min \left\{\frac{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}}}{m^{\frac{4}{5}}},\frac{|B||C{|}^{\frac{1}{2}}}{m},\frac{|B||A{|}^{\frac{1}{2}}}{m},\frac{|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}}{m^{\frac{2}{3}}}\right\}.$$

$$\max \left\{|f(A,A)|,|A\cdot A|\right\}\gg |A{|}^{\frac{6}{5}}.$$

$$\max \left\{|A^{-1}+B|,|B\cdot C|\right\}\gg \min \left\{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}},|B||C{|}^{\frac{1}{2}},|B||A{|}^{\frac{1}{2}},|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}\right\}.$$

$$\max \left\{|A(B+1)|,|B\cdot C|\right\}\gg \min \left\{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}},|B||C{|}^{\frac{1}{2}},|B||A{|}^{\frac{1}{2}},|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}\right\}.$$

$$\max \left\{|f(A,B)|,|B+C|\right\}\gg \min \left\{\frac{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}}}{m^{\frac{4}{5}}},\frac{|B||C{|}^{\frac{1}{2}}}{m},\frac{|B||A{|}^{\frac{1}{2}}}{m},\frac{|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}}{m^{\frac{2}{3}}}\right\}.$$

$$\max \left\{|f(A,A)|,|A+A|\right\}\gg |A{|}^{\frac{6}{5}}.$$

$$\max \left\{|A(B+1)|,|B+C|\right\}\gg \min \left\{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}},|B||C{|}^{\frac{1}{2}},|B||A{|}^{\frac{1}{2}},|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}\right\}.$$

$$\max \left\{|A\cdot B|,|B+C|\right\}\gg \min \left\{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}},|B||C{|}^{\frac{1}{2}},|B||A{|}^{\frac{1}{2}},|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}\right\}$$

$$\min \left\{|A+A|,|A\cdot A|\right\}\le |A{|}^{\frac{9}{8}-ϵ}$$

$$|f(A,A)|\gg |A{|}^{\frac{5}{4}+\frac{2ϵ}{3}}.$$

$$\mathcal{I}(\mathcal{R},\mathcal{S})\ll |\mathcal{R}{|}^{1/2}|\mathcal{S}|+k|\mathcal{S}|.$$

$$E_{\lambda }=\left|\left\{\left(f(a,b),c\cdot g(a{)}^{-1},c\cdot h(a)\right):(a,b,c)\in A\times B\times C,f(a,b)\cdot c\cdot g(a{)}^{-1}-c\cdot h(a)=\lambda \right\}\right|,$$

$$g(a)(h(a)+b)=x,c\cdot g(a{)}^{-1}=y,c\cdot h(a)=z.$$

$$g(a)h(a)=z{y}^{-1}.$$

$$|A||B||C|/m\le \sum _{\lambda \in B\cdot C}{E}_{\lambda }.$$

$${\left(|A||B||C|/m\right)}^2\le {\left(\sum _{\lambda \in B\cdot C}{E}_{\lambda }\right)}^2\le E\cdot |B\cdot C|,$$

$$\mathcal{R}=\left\{\left(c\cdot g(a{)}^{-1},c\cdot h(a),g(a^{\prime })(h(a^{\prime })+b^{\prime })\right):a,a^{\prime }\in A,b^{\prime }\in B,c\in C\right\}$$

$$\mathcal{S}=\left\{g(a)(h(a)+b)X-Y-c^{\prime }g(a^{\prime }{)}^{-1}Z=-c^{\prime }\cdot h(a^{\prime }):a,a^{\prime }\in A,b\in B,c^{\prime }\in C\right\}.$$

$$k\le \max \{|A|,|C|,|f(A,B)|\}.$$

$$I(\mathcal{R},\mathcal{S})\le |f(A,B){|}^{3/2}|A{|}^{3/2}|C{|}^{3/2}+k|f(A,B)|A||C|.$$

$$\max \left\{|f(A,B)|,|B\cdot C|\right\}\gg \min \left\{\frac{|A{|}^{\frac{1}{5}}|B{|}^{\frac{4}{5}}|C{|}^{\frac{1}{5}}}{m^{\frac{4}{5}}},\frac{|B||C{|}^{\frac{1}{2}}}{m},\frac{|B||A{|}^{\frac{1}{2}}}{m},\frac{|B{|}^{\frac{2}{3}}|C{|}^{\frac{1}{3}}|A{|}^{\frac{1}{3}}}{m^{\frac{2}{3}}}\right\}.$$

Full text

1 Introduction

Throughout this chapter, by $\mathbb{F}$ we refer to any arbitrary field, while by $\mathbb{F}_{p}$, we only refer to the fields of prime order $p$. We denote the set of non-zero elements by $\mathbb{F}^{*}$ and $\mathbb{F}^{*}_{p}$, respectively. Furthermore, we use the following convention: if the characteristic of $\mathbb{F}$ is positive, then we denote its characteristic by $p$; if the characteristic of $\mathbb{F}$ is zero, then we set $p=\infty$. So a term like $N<p^{5/8}$ is restrictive in positive characteristic, but vacuous for zero one.

For $A\subset\mathbb{F}$, the sum and the product sets are defined as follows:

[TABLE]

For $A\subset\mathbb{F}_{p}$, Bourgain, Katz and Tao ([2]) proved that if $p^{\delta}<|A|<p^{1-\delta}$ for some $\delta>0$, then we have

[TABLE]

for some $\epsilon=\epsilon(\delta)>0$. Here, and throughout, by $X\ll Y$ we mean that there exists the constant $C>0$ such that $X\leq CY$.

In a breakthrough paper [8], Roche-Newton, Rudnev, and Shkredov improved and generalized this result to arbitrary fields. More precisely, they showed that for $A\subset\mathbb{F}$, the sum set and the product set satisfy

[TABLE]

We note that the same bound also holds for $|A(1+A)|$ [11], and $|A+A^{2}|$, $\max\left\{|A+A|,|A^{2}+A^{2}|\right\}$ [7]. We refer the reader to [1, 3, 8, 6] and references therein for recent results on the sum-product topic.

Let $G$ be a subgroup of $\mathbb{F}^{*}$, and $g\colon G\to\mathbb{F}^{*}$ be an arbitrary function. We define

[TABLE]

For $A,B\subset\mathbb{F}_{p}$ and two-variable functions $f(x,y)$ and $g(x,y)$ in $\mathbb{F}_{p}[x,y]$, Hegyvári and Hennecart [4], using graph theoretic techniques, proved that if $|A|=|B|=p^{\alpha}$, then

[TABLE]

for some $\Delta(\alpha)>0$. More precisely, they established the following results.

Theorem 1.1** (Hegyvári and Hennecart, [4]).**

Let $G$ be a subgroup of $\mathbb{F}_{p}^{*}$. Consider the function $f(x,y)=g(x)(h(x)+y)$ on $G\times\mathbb{F}_{p}^{*}$, where $g,h\colon G\to\mathbb{F}_{p}^{*}$ are arbitrary functions. Define $m=\mu(g\cdot h)$. For any subsets $A\subset G$ and $B,C\subset\mathbb{F}_{p}^{*}$, we have

[TABLE]

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

Let $G$ be a subgroup of $\mathbb{F}_{p}^{*}$. Consider the function $f(x,y)=g(x)(h(x)+y)$ on $G\times\mathbb{F}_{p}^{*}$, where $g,h\colon G\to\mathbb{F}_{p}^{*}$ are arbitrary functions. Define $m=\mu(g)$. For any subsets $A\subset G$, $B,C\subset\mathbb{F}_{p}^{*}$, we have

[TABLE]

It is worth noting that Theorem $6$ established by Bukh and Tsimerman [3] does not cover such a function defined in Theorem 1.2. The reader can also find the generalizations of Theorems 1.1 and 1.2 in the setting of finite valuation rings in [5].

Suppose $f(x,y)=g(x)(h(x)+y)$ with $\mu(g),\mu(h)=O(1)$ and $A=B=C$. Then, it follows from Theorems 1.1 and 1.2 that

If $|A|\gg p^{2/3}$, then we have

[TABLE] 2. 2.

If $|A|\ll p^{2/3}$, then we have

[TABLE]

The main goal of this paper is to improve and generalize Theorems 1.1 and 1.2 to arbitrary fields for small sets. Our first result is an improvement of Theorem 1.1.

Theorem 1.3**.**

Let $f(x,y)=g(x)(h(x)+y)$ be a function defined on $\mathbb{F}^{*}\times\mathbb{F}^{*}$, where $g,h\colon\mathbb{F}^{*}\to\mathbb{F}^{*}$ are arbitrary functions. Define $m=\mu(g\cdot h)$. For any subsets $A,B,C\subset\mathbb{F}^{*}$ with $|A|,|B|,|C|\leq p^{5/8}$, we have

[TABLE]

The following are consequences of Theorem 1.3.

Corollary 1.4**.**

Let $f(x,y)=g(x)(h(x)+y)$ be a function defined on $\mathbb{F}^{*}\times\mathbb{F}^{*}$, where $g,h\colon\mathbb{F}^{*}\to\mathbb{F}^{*}$ are arbitrary functions with $\mu(g\cdot h)=O(1)$. For any subset $A\subset\mathbb{F}^{*}$ with $|A|\leq p^{5/8}$, we have

[TABLE]

Corollary 1.5**.**

Consider the subsets $A\subset\mathbb{F}^{*}$, and $B,C\subset\mathbb{F}$ with $|A|,|B|,|C|\leq p^{5/8}$.

By fixing $g(x)=1$ and $h(x)=x^{-1}$, we get

[TABLE] 2. 2.

By fixing $g(x)=x$ and $h(x)=1$, we have

[TABLE]

It follows from Corollary 1.5(2) that if $B=A$ and $C=A+1$ then we have $|A(A+1)|\gg|A|^{6/5}$, which recovers the result of Stevens and de Zeeuw [11].

Our next result is the additive version of Theorem 1.3, which improves Theorem 1.2.

Theorem 1.6**.**

Let $f(x,y)=g(x)(h(x)+y)$ be a function defined on $\mathbb{F}^{*}\times\mathbb{F}^{*}$, where $g\colon\mathbb{F}^{*}\to\mathbb{F}^{*}$ are arbitrary functions. Define $m=\mu(g)$. For any subsets $A,B,C\subset\mathbb{F}^{*}$ with $|A|,|B|,|C|\leq p^{5/8}$, we have

[TABLE]

Corollary 1.7**.**

Let $f(x,y)=g(x)(h(x)+y)$ be a function defined on $\mathbb{F}^{*}\times\mathbb{F}^{*}$, where $g\colon\mathbb{F}^{*}\to\mathbb{F}^{*}$ are arbitrary functions with $\mu(g)=O(1)$. For any subset $A\subset\mathbb{F}^{*}$ with $|A|\leq p^{5/8}$, we have

[TABLE]

Let $g(x)=x$ and $h(x)=1$, we have the following corollary.

Corollary 1.8**.**

For $A,B,C\subset\mathbb{F}$ with $|A|,|B|,|C|\leq p^{5/8}$, we have

[TABLE]

By fixing $g(x)=x$ and $h(x)=0$, we have the following result.

Corollary 1.9**.**

For $A,B,C\subset\mathbb{F}$ with $|A|,|B|,|C|\ll p^{5/8}$, we have

[TABLE]

In the case $A=B=C$, we recover the following result due to Roche-Newton, Rudnev, and Shkredov [8], which says that $\max\left\{|A+A|,|A\cdot A|\right\}\gg|A|^{6/5}$.

It has been shown in [11] that if $f(x,y)=x(x+y)$, then $|f(A,A)|\gg|A|^{5/4}$ under the condtion $|A|\leq p^{2/3}$. In the following theorem, we show that if either $|A+A|$ or $|A\cdot A|$ is sufficiently small, the exponent $5/4$ can be improved from the polynomials to a larger family of functions on $\mathbb{F}^{*}\times\mathbb{F}^{*}$

Theorem 1.10**.**

Let $f(x,y)=g(x)(h(x)+y)$ be a function defined on $\mathbb{F}^{*}\times\mathbb{F}^{*}$, where $g,h\colon\mathbb{F}^{*}\to\mathbb{F}^{*}$ are arbitrary functions with $\mu(f),\mu(g)=O(1)$. Consider the subset $A\subset\mathbb{F}^{*}$ with $|A|\leq p^{5/8}$, satisfying

[TABLE]

for some $\epsilon>0$. Then, we have

[TABLE]

2 Proofs of Theorems 1.3, 1.6, and 1.10

Let $\mathcal{R}$ be a set of points in $\mathbb{F}^{3}$ and $\mathcal{S}$ be a set of planes in $\mathbb{F}^{3}$. We write $\mathcal{I}(\mathcal{R},\mathcal{S})=|\{(r,s)\in\mathcal{R}\times\mathcal{S}:r\in s\}|$ for the number of incidences between $\mathcal{R}$ and $\mathcal{S}$. To prove Theorems 1.3 and 1.6, we make use of the following point-plane incidence bound due to Rudnev [10]. A short proof can be found in [12].

Theorem 2.1** (Rudnev, [10]).**

Let $\mathcal{R}$ be a set of points in $\mathbb{F}^{3}$ and let $\mathcal{S}$ be a set of planes in $\mathbb{F}^{3}$, with $|\mathcal{R}|\ll|\mathcal{S}|$ and $|\mathcal{R}|\ll p^{2}$. Assume that there is no line containing $k$ points of $\mathcal{R}$. Then

[TABLE]

Proof of Theorem 1.3:

Define $f(A,B)=\{f(a,b)\colon a\in A,b\in B\},g(A)=\{g(a)\colon a\in A\}$, $h(A)=\{h(a)\colon a\in A\}$. For $\lambda\in B\cdot C$, let

[TABLE]

where by $g(a)^{-1}$ we mean the multiplicative inverse of $g(a)$ in $\mathbb{F}^{*}$. For a given triple $(x,y,z)\in(\mathbb{F}^{*})^{3}$, we count the number of solutions $(a,b,c)\in A\times B\times C$ to the following system

[TABLE]

This implies that

[TABLE]

Since $\mu(g\cdot h)=m$, there are at most $m$ different values of $a$ satisfying the equation $g(a)h(a)=zy^{-1}$, and $b,c$ are uniquely determined in term of $a$ by the first and second equations of the system. This implies that

[TABLE]

By the Cauchy-Schwarz inequality, we get

[TABLE]

where $E=\sum_{\lambda\in B\cdot C}E_{\lambda}^{2}$.

Define the point set $\mathcal{R}$ as

[TABLE]

and the set of planes $\mathcal{S}$ as

[TABLE]

We have $E\leq I(\mathcal{R},\mathcal{S})$, and $|\mathcal{R}|=|\mathcal{S}|\leq|f(A,B)||A||C|$. To apply Theorem 2.1, we need to find an upper bound on $k$ which is the maximum number of collinear points in $\mathcal{R}$. The projection of $\mathcal{R}$ into the first two coordinates is the set $\mathcal{T}=\{(c\cdot g(a)^{-1},c\cdot h(a))\colon a\in A,c\in C\}$. The set $\mathcal{T}$ can be covered by the lines of the form $y=g(a)h(a)x$ with $a\in A$. This implies that $\mathcal{T}$ can be covered by at most $|A|$ lines passing through the origin, with each line containing $|C|$ points of $\mathcal{T}$. Therefore, a line in $\mathbb{F}^{3}$ contains at most $\max\{|A|,|C|\}$ points of $\mathcal{R}$, unless it is vertical, in which case it contains at most $|f(A,B)|$ points. In other words, we get

[TABLE]

If $|\mathcal{R}|\gg p^{2}$, then we get $|f(A,B)||A||C|\gg p^{2}$. Since $|A|,|C|\leq p^{5/8}$, we have $|f(A,B)|\gg p^{3/4}\gg|A|^{\frac{1}{5}}|B|^{\frac{4}{5}}|C|^{\frac{1}{5}}$, and we are done in this case. Thus, we can assume that $|\mathcal{R}|\ll p^{2}$. Applying Theorem 2.1, we obtain

[TABLE]

Putting (2) and (3) together gives us

[TABLE]

This completes the proof of the theorem. $\square$

Proof of Theorem 1.6:

The proof goes in the same direction as Theorem 1.3, but for the sake of completeness, we include the detailed proof. For $\lambda\in B+C$, let

[TABLE]

For a given triple $(x,y,z)\in(\mathbb{F}^{*})^{3}$, we count the number of solutions $(a,b,c)\in A\times B\times C$ to the following system

[TABLE]

Since $\mu(g)=m$, there are at most $m$ different values of $a$ satisfying the equation $g(a)=y^{-1}$, and $b,c$ are uniquely determined in term of $a$ by the first and third equations of the system. This implies that

[TABLE]

By the Cauchy-Schwarz inequality, we have

[TABLE]

where $E=\sum_{\lambda\in B+C}E_{\lambda}^{2}$. Define the point set $\mathcal{R}$ as

[TABLE]

and the collection of planes $\mathcal{S}$ as

[TABLE]

It is clear that $|\mathcal{R}|=|\mathcal{S}|\leq|f(A,B)||A||C|$, and $E\leq I(\mathcal{R},\mathcal{S})$. To apply Theorem 2.1, we need to find an upper bound on $k$ which is the maximum number of collinear points in $\mathcal{R}$. The projection of $\mathcal{R}$ into the first two coordinates is the set $\mathcal{T}=\{\left(g(a)^{-1},c-h(a)\right)\colon a\in A,c\in C\}$. The set $\mathcal{T}$ can be covered by at most $|A|$ lines of the form $x=g(a)^{-1}$ with $a\in A$, where each line contains $|C|$ points of $\mathcal{T}$. Therefore, a line in $\mathbb{F}^{3}$ contains at most $\max\{|A|,|C|\}$ points of $\mathcal{R}$, unless it is vertical, in which case it contains at most $|f(A,B)|$ points. So we get

[TABLE]

If $|\mathcal{R}|\gg p^{2}$, this implies that $|f(A,B)||A||C|\gg p^{2}$. Since $|A|,|C|\leq p^{5/8}$, we have $|f(A,B)|\gg p^{3/4}\gg|A|^{\frac{1}{5}}|B|^{\frac{4}{5}}|C|^{\frac{1}{5}}$, and we are done. Thus, we can assume that $|\mathcal{R}|\ll p^{2}$. Applying Theorem 2.1, we obtain

[TABLE]

Putting (4) and (5) together gives us

[TABLE]

This completes the proof. $\square$

Proof of Theorem 1.10:

One can assume that $|f(A,A)|\leq|A|^{2}$, since otherwise we are done. Now by the proofs of Theorems 1.3 and 1.6 for $A\subset\mathbb{F}^{*}$ with $|A|\leq p^{5/8}$, we have

[TABLE]

Since $\min\left\{|A+A|,|A\cdot A|\right\}\leq|A|^{\frac{9}{8}-\epsilon}$, we get $|f(A,A)|^{3/2}\gg|A|^{3-\frac{9}{8}+\epsilon}$, which concludes the proof of the theorem. $\square$

3 Acknowledgments

The authors were partially supported by Swiss National Science Foundation grants 200020-162884 and 200021-175977.