On the size and structure of $t$-representable sumsets

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