How iterated sumsets grow under the assumption of small doubling

Abstract

Let $K>1$ be a real number and let $A\subsetneq \mathbb Z$ be a finite and nonempty subset. Suppose $|A+A|<K|A|$. A fundamental open problem is to characterize the size of higher iterated sumsets $nA = A+\cdots+A$ ($n$-fold sumset). In 2004, Terry Tao formally posed the problem of determining the optimal growth function governing iterated sumsets of small-doubling sets. In this manuscript, we give a solution to this long-standing open problem for large enough $n$.