Abstract
Let A⊆Z≥0 be a finite set with minimum element 0, maximum element m, and ℓ 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” forh≥(1+o(1))1emℓt1/ℓ (as ℓ→∞, t1/ℓ→∞), and prove a similar theorem on the size and structure of A⊆Zd for h sufficiently large. Moreover, we construct a family of sets A=A(m,ℓ,t)⊆Z≥0 for which (hA)(t) is not structured for h≪mℓt1/ℓ.
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