Expanding polynomials for sets with additive structure

The expansion of bivariate polynomials is well-understood for sets with a linear-sized product set. In contrast, not much is known for sets with small sumset. In this work, we provide expansion bounds for polynomials of the form $f(x,y)=g(x+p(y))+h(y)$ for sets with small sumset. In particular, we prove that when $\mathrm{\mid }A\mathrm{\mid }$, $\mathrm{\mid }B\mathrm{\mid }$, $\mathrm{\mid }A+A\mathrm{\mid }$, and $\mathrm{\mid }B+B\mathrm{\mid }$ are not too far apart, for every $\epsilon >0$ we have $$\mathrm{\mid }f(A,B)\mathrm{\mid }=\mathrm{\Omega }\left(\frac{\mathrm{\mid }A{\mathrm{\mid }}^{256\mathrm{/}121-\epsilon }\mathrm{\mid }B{\mathrm{\mid }}^{74\mathrm{/}121-\epsilon }}{\mathrm{\mid }A+A{\mathrm{\mid }}^{108\mathrm{/}121}\mathrm{\mid }B+B{\mathrm{\mid }}^{24\mathrm{/}121}}\right)\mathrm{.}$$ We show that the above bound and its variants have a variety of applications in additive combinatorics and distinct distances problems. Our proof technique relies on the recent proximity approach of Solymosi and Zahl. In particular, we show how to incorporate the size of a sumset into this approach.