How iterated sumsets grow under the assumption of small doubling

Let $K>1$ be a real number and let $A⊊\mathbb{Z}$ be a finite and nonempty subset. Suppose $\mathrm{\mid }A+A\mathrm{\mid }<K\mathrm{\mid }A\mathrm{\mid }$. 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 $n$.