An improved bound on the sum-product estimate in $\mathbb{F}_{p}$

For a subset A of a field F, write A(A + 1) for the set {a(b + 1):a,b\in A}. We establish new estimates on the size of A(A+1) in the case where F is either a finite field of prime order, or the real line. In the finite field case we show that A(A+1) is of cardinality at least C|A|^{57/56-o(1)} for some absolute constant C, so long as |A| < p^{1/2}. In the real case we show that the cardinality is at least C|A|^{24/19-o(1)}. These improve on the previously best-known exponen...

Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, \qquad a_{1},\ldots,a_{k} \in A,$$ with $|S| \geq 2^{-12}p^{-\epsilon}\min\{|A||B|,p\}$. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $\alpha < 1$ and $\beta,\delta > 0$, there ...

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...

In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s dist...

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...

Assume that $A\subseteq \Fp, B\subseteq \Fp^{*}$, $\1/4\leqslant\frac{|B|}{|A|},$ $|A|=p^{\alpha}, |B|=p^{\beta}$. We will prove that for $p\geqslant p_0(\beta)$ one has $$\sum_{b\in B}E_{+}(A, bA)\leqslant 15 p^{-\frac{\min\{\beta, 1-\alpha\}}{308}}|A|^3|B|.$$ Here $E_{+}(A, bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.

In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}-o(1)} \] holds for all $A \subset \mathbb R$, and for all convex functions $f$ which satisfy an additional technical condition. This technical condition is satisfied by the logarithmic function, and this fact can be used to deduce a sum-product estimate \[ \max \{ |16A| , |A^{(16)}| \} \gg |A|^{\frac{3}{2} + c}, \] for some $c>0$. Previously, no sum-product estimate ...

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.

In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In particular, we show that for any finite set $A\subset{\mathbb{R}}$ and any strictly convex or concave function $f$, \[|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}\] and \[\max\{|A-A|,\ |f(A)+f(A)|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}....

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...