Slightly improved sum-product estimates in fields of prime order

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 $p$.

Let $F_p$ be the field of a prime order $p.$ For a subset $A\subset F_p$ we consider the product set $A(A+1).$ This set is an image of $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p.$ In the present paper we show that if $|A|<p^{1/2},$ then $$ |A(A+1)|\ge |A|^{106/105+o(1)}.$$ If $|A|>p^{2/3},$ then we prove that $$|A(A+1)|\gg \sqrt{p |A|}$$ and show that this is the optimal in general settings bound up to the implied constant. We also estimate the...

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A = A^*$, then $|A| < p/9 + o(p)$. $(iii)$ If $|A| \gg \frac{\log\log{p}}{\sqrt{\log{p}}}p$, then $|A + A^*| \geqslant (1 - o(1))\min(2\sqrt{|A|p}, p)$. Here the constants $1/8$, $1/9$, and $2$ are the best possible. The proof involves \em...

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper, answer Szemer\'edi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\epsilon}$ for some $\epsilon>0$. In fact, this...

We generalize two results about subgroups of multiplicative group of finite field of prime order. In particular, the lower bound on the cardinality of the set of values of polynomial $P(x,y)$ is obtained under the certain conditions, if variables $x$ and $y$ belong to a subgroup $G$ of the multiplicative group of the filed of residues. Also the paper contains a proof of the result that states that if a subgroup $G$ can be presented as a set of values of the polynomial $P(x,y)...

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max\{|AB|, |A+C| \}$ and $\max\{|(A+\alpha)B|, |A+C|\}$, $\alpha \neq 0$ in the regime when the sizes of finite subsets $A,B,C$ of a field differ significantly.

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. More precisely, we will prove that 1. If $|\mathcal A|\gg q^{r-\frac{1}{3}}$, then $$\max\left\lbrace |\mathcal A+\mathcal A|, |\mathcal A^2+\mathcal A^2|\right\rbrace \gg q^{\frac{r}{2}}|\mathcal A|^{\frac{1}{2}}.$$ 2. If $q^{r-\frac{3}{8}}...

Let $F$ be a field of characteristic $p>2$ and $A\subset F$ have sufficiently small cardinality in terms of $p$. We improve the state of the art of a variety of sum-product type inequalities. In particular, we prove that $$ |AA|^2|A+A|^3 \gg |A|^6,\qquad |A(A+A)|\gg |A|^{3/2}. $$ We also prove several two-variable extractor estimates: ${\displaystyle |A(A+1)| \gg|A|^{9/8},}$ $$ |A+A^2|\gg |A|^{11/10},\; |A+A^3|\gg |A|^{29/28}, \; |A+1/A|\gg |A|^{31/30}.$$ Besides, we addres...

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 subfield. An analogous result was established in \cite{LiORN}, with the product set $A\cdot{A}$ in the place of the ratio set $A:A$. The sum-ratio estimate here beats the sum-product estimate in \cite{LiORN} by a logarithmic factor, with s...

There exists an absolute constant $C$ with the following property. Let $A \subseteq \mathbb{F}_p$ be a set in the prime order finite field with $p$ elements. Suppose that $|A| > C p^{5/8}$. The set \[ (A \pm A)(A \pm A) = \{(a_1 \pm a_2)(a_3 \pm a_4) : a_1,a_2,a_3,a_4 \in A\} \] contains at least $p/2$ elements.