A note on expansion in prime fields

Abstract

Let $\beta ,\epsilon \in (0,1]$, and $k\ge \exp (122\max \{1/\beta ,1/\epsilon \})$. We prove that if $A,B$ are subsets of a prime field ${\mathbb{Z}}_p$, and $|B|\ge p^{\beta }$, then there exists a sum of the form $$S=a_{1}B\pm \dots \pm a_{k}B,a_1,\dots ,a_k\in A,$$ with $|S|\ge 2^{-12}{p}^{-\epsilon }\min \{|A||B|,p\}$. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $\alpha <1$ and $\beta ,\delta >0$, there exists $\epsilon >0$ such that the following holds. If $A,B,E\subset {\mathbb{Z}}_p$ satisfy $|A|\le p^{\alpha }$, $|B|\ge p^{\beta }$, and $|B||E|\ge p^{\delta }|A|$, then there exists $t\in E$ such that $$|A+tB|\ge c{p}^{\epsilon }|A|,$$ for some absolute constant $c>0$. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.

7 pages

References in corpus (1)

• Sum-product phenomena in F_p: a brief introduction