When does the equality hold in Dias da Silva - Hamidoune Theorem?

Question

Let $p$ be prime number and let $A$ be a $k$-elements subset of $\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that $|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where $h$ is an integer with $2 \leq h \leq k$ and $h^{\hat{}}A$ denote the set of all sums of $h$ distinct elements of $A$. What is/are the characterization of the set $A$ for which $|h^{\hat{}}A|$ attains the lower bound ? In other words, is the lower bound optimal in the above theorem ?

I know that the equality occurs for arithmetic progressions in case of $h = 2$.

Answer 1

The lower bound is optimal, and the bound is attained on arithmetic progressions: say, for $A=[1,k]$ one has $h^{\hat{}}A=[h(h+1)/2,kh-h(h-1)/2]$ whence $|h^{\hat{}}A|=kh-h^2+1$ (provided, say, $kh<p$).

Characterizing those $A$ for which equality is attained is subtler. In the case where $A$ is a set of integers, Nathanson has shown that if $k:=|A|\ge 5$, and $2\le h\le k-2$, then $|h^{\hat{}}A|=hk-h^2+1$ is only possible if $A$ is an arithmetic progression. I am not sure as to whether a similar result for the prime-order groups is known.