A sum-product estimate in fields of prime order

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present paper we prove that if $A\subset \mathbb{F}_p$ with $|A|>p^{2/3},$ then $$ \max\{|A+A|, |AA|\}\gg p^{1/2}|A|^{1/2}. $$

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if $|A|\succeq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succapprox \min\{|A|^{13/12}(\frac{|A|}{p^{0.5}})^{1/12},|A|(\frac{p}{|A|})^{1/11}\}.\] These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on differ...

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a non-empty subset of $\mathbb{F}_p.$ In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain, Glibichuk, Konyagin on the size of $\max\{|A+A|, |AA|\}.$ In particular, our result implies that if $1<|A|\le p^{7/13}(\log p)^{-4/13},$ then $$ \max\{|A+A|, |AA|\}\gg \frac{|A|^{15/14}}{(\log|A|)^{2/7}} . $$

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 $\max \{|A+A|, |A \cdot A|\} \gtrsim \frac{{|A|}^{{3/2}}}{q^{{1/4}}$.

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 estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev (\cite{mishaSP}).

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 $|AA+A|,|A(A+A)|\gg q$.

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: $$ \max \{|A+A|,|A A|\} \gg \min \left\{\frac{|A|^{15 / 14} \max \left\{1,|A|^{1 / 7} p^{-1 / 14}\right\}}{(\log |A|)^{2 / 7}}, \frac{|A|^{11 / 12} p^{1 / 12}}{(\log |A|)^{1 / 3}}\right\}, $$ and more importantly: $$\max \{|A+A|,|A A|\} \gg \frac{|A|^{15 / 14}}{(\log |A|)^{2 ...

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 is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as...

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]

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 $\max\{|A+A|,|AA|\}\geq C |A|^{6/5-\epsilon}$. In particular such a result is obtained for the rational function field $\mathbb{F}_q(t)$. Identical results are also obtained for finite subsets of the $p$-adic field $\mathbb{Q}_p$ for any prime $p$.