The Cauchy-Davenport Theorem says that if $A_1, \ldots, A_k$ are subsets of ${\mathbb Z}_p$, $p$ prime, then $| \sum_i A_i | \geq \min (p, \sum_i |A_i| -k +1)$.

I am looking for a generalization that bounds the number of ways each element $a \in \sum_i A_i$ can be represented as $a=\sum_i a_i$ with $a_i \in A_i$.

Specifically, I am interested in the case where $\sum_i |A_i| -k +1 \geq p$. Let $N_{min}$ denote the minimum number of ways any element can be written as a sum, and let $N_{max}$ denote the maximum number of ways any element can be written as a sum. Can we bound the ratio $N_{max}/N_{min}$? My vague conjecture would be that for $\sum_i |A_i| -k +1 \gg p$ we can bound the ratio by a constant, but I have no progress toward a proof.

I am aware of two papers by Pollard from the '70s and some followup work looking at sort-of-related questions, but am not aware of anything that comes close to addressing the above.

On complexes in a semigroup, Indag. Math., Vol. 18 (1956), 247-254. – Salvo Tringali Jan 18 '13 at 11:16