An explicit sum-product estimate in $\mathbb{F}_p$

The classical Cauchy--Davenport inequality gives a lower bound for the size of the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to prove a considerable strengthening of this inequality, where we take only a small number of points from each of the two subsets when forming the sum. One of our results is that there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then there...

In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $\Delta(\alpha)>0$ such that if $|A|\asymp p^{\alpha}$ then $$ \max(|A+A|,|A\cdot A|)\gg |A|^{1+\Delta(\alpha)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g...

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...

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 \pi i z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of conse...

In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In addition, several generalizations of these results are given. In the case that $A$ is a subset of a prime field $\mathbb F_p$ of size less than $p^{1/2}$ it is shown that $|\{a^2+b:a,b \in A\}|\geq C |A|^{147/146}$, where $|\cdot|$ denotes the cardinality of the set and $C$ is an abso...

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.

We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and ...

Let $A$ be a multiplicative subgroup of $\mathbb Z_p^*$. Define the $k$-fold sumset of $A$ to be $kA=\{x_1+\dots+x_k:x_i \in A,1\leq i\leq k\}$. We show that $6A\supseteq \mathbb Z_p^*$ for $|A| > p^{\frac {11}{23} +\epsilon}$. In addition, we extend a result of Shkredov to show that $|2A|\gg |A|^{\frac 85-\epsilon}$ for $|A|\ll p^{\frac 59}$.

In this paper we present a procedure which allows to transform a subset $A$ of $\mathbb{Z}_{p}$ into a set $ A'$ such that $ |2\hspace{0.15cm}\widehat{} A'|\leq|2\hspace{0.15cm}\widehat{} A | $, where $2\hspace{0.15cm}\widehat{} A$ is defined to be the set $\left\{a+b:a\neq b,\;a,b\in A\right\}$. From this result, we get some lower bounds for $ |2\hspace{0.15cm}\widehat{} A| $. Finally, we give some remarks related to the problem for which sets $A\subset \mathbb{Z}_{p}$ we ha...

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i...