Higher convexity and iterated sum sets

Abstract

Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\le K|A|$, $|2^{k}f(A)-(2^k-1)f(A)|{\gg }_k|A{|}^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type implications, e.g. that $A+A$ being small implies unbounded growth for a many enough times iterated product set $A\cdots A$.