On the minimum size of restricted sumsets in cyclic groups

TL;DR

This paper establishes an explicit upper bound for the minimum size of restricted sumsets in cyclic groups for all parameters, confirming known cases and providing counterexamples to existing conjectures.

Contribution

It introduces a universal upper bound for restricted sumsets in cyclic groups, extending understanding beyond prime cases and challenging previous conjectures.

Findings

Bound is tight for all known cases up to n=40

Counterexamples found for some existing conjectures

Bound applies to all n, m, h values

Abstract

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.