A note on expansion in prime fields

Let $\beta ,ϵ\in (0,1]$, and $k \geq \exp(122 \max{1/\beta,1/\epsilon})$\mathrm{.}Weprovethatif$A,B$ are subsets of a prime field ${\mathbb{Z}}_p$, and $\mathrm{\mid }B\mathrm{\mid }\ge p^{\beta }$, 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 $\mathrm{\mid }S\mathrm{\mid }\ge 2^{-12}{p}^{-ϵ}\min \{\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid },p\}$. 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$suchthatthefollowingholds\mathrm{.}If$A,B,E \subset $\mathbb{Z}$_{p}$satisfy$|A| \leq p^{\alpha}$,$|B| \geq p^{\beta}$,and$|B||E| \geq p^{\delta}|A|$, then there exists $t\in E$ such that $$\mathrm{\mid }A+tB\mathrm{\mid }\ge c{p}^{ϵ}\mathrm{\mid }A\mathrm{\mid },$$ for some absolute constant $c>0$. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.