A note on expansion in prime fields

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 exists $\epsilon > 0$ such that the following holds. 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 $$|A + tB| \geq c p^{\epsilon}|A|,$$ for some absolute constant $c > 0$. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.