On the size and structure of t-representable sumsets

Let $A\subseteq Z_{\ge 0}$ be a finite set with minimum element 0, maximum element m, and ℓ elements strictly in between. Write ${\left(hA\right)}^{\left(t\right)}$ for the set of integers that can be written in at least t ways as a sum of h elements of A. We prove that ${\left(hA\right)}^{\left(t\right)}$ is “structured” for$h\ge (1+o(1\left)\right)\frac{1}{e}m\ell t^{1/\ell }$ (as $\ell \to \infty$, $t^{1/\ell }\to \infty$), and prove a similar theorem on the size and structure of $A\subseteq Z^d$ for h sufficiently large. Moreover, we construct a family of sets $A=A(m,\ell ,t)\subseteq Z_{\ge 0}$ for which ${\left(hA\right)}^{\left(t\right)}$ is not structured for $h\ll m\ell t^{1/\ell }$.