Related papers: Sum-product estimates over arbitrary finite fields

This paper gives an improved sum-product estimate for subsets of a finite field whose order is not prime. It is shown, under certain conditions, that $$\max\{|A+A|,|A\cdot{A}|\}\gg{\frac{|A|^{12/11}}{(\log_2|A|)^{5/11}}}.$$ This new…

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…

Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$.…

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a…

We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field…

The aim of this note is to record a proof that the estimate $$\max{\{|A+A|,|A:A|\}}\gg{|A|^{12/11}}$$ holds for any set $A\subset{\mathbb{F}_q}$, provided that $A$ satisfies certain conditions which state that it is not too close to being a…

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim |A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim…

Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge c|A|^{1+\epsilon}$ for some $\epsilon>0$ and c>0. This extends the result of J. Bourgain, N. Katz, and…

We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q^{n^2}, $$ whenever $|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$…

We improve the best known sum-product estimates over the reals. We prove that \[ \max(|A+A|,|AA|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,, \] for a finite $A\subset \mathbb R$, following a streamlining of the arguments of Solymosi,…

Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this paper, we improve a recent result due to Yazici (2018) on a sum-product type problem.…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is…

We give an improved bound on the famed sum-product estimate in a field of residue class modulo $p$ ($\mathbb{F}_{p}$) by Erd\H{o}s and Szemeredi, and a non-empty set $A \subset \mathbb{F}_{p}$ such that: $$ \max \{|A+A|,|A A|\} \gg \min…

Let $R$ be a finite valuation ring of order $q^r.$ Using a point-plane incidence estimate in $R^3$, we obtain sum-product type estimates for subsets of $R$. In particular, we prove that for $A\subset R$, $$|AA+A|\gg \min\left\{q^{r},…

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also…

We prove that if $A \subset {\Bbb F}_q$ is such that $$|A|>q^{{1/2}+\frac{1}{2d}},$$ then $${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$ where $$A^2=\{a \cdot a': a,a' \in A\},$$ and where ${\Bbb F}_q^{*}$ denotes the…

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that…

We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $\delta=|A|^{-\alpha}$, we…