On the minimum size of restricted sumsets in cyclic groups

For positive integers $n$, $m$, and $h$, we let $\rho \hat{\;}(\mathbb{Z}_n, m, h)$ denote the minimum size of the $h$-fold restricted sumset among all $m$-subsets of the cyclic group of order $n$. The value of $\rho \hat{\;}(\mathbb{Z}_n, m, h)$ was conjectured for prime values of $n$ and $h=2$ by Erd\H{o}s and Heilbronn in the 1960s; Dias da Silva and Hamidoune proved the conjecture in 1994 and generalized it for an arbitrary $h$, but little is known about the case when $n$ is composite. Here we exhibit an explicit upper bound for all $n$, $m$, and $h$; our bound is tight for all known cases (including all $n$, $m$, and $h$ with $n \leq 40$). We also provide counterexamples for conjectures made by Plagne and by Hamidoune, Llad\'o, and Serra.