Multiplicative subgroups are not restricted sumsets

Abstract

We determine exactly which proper multiplicative subgroups of a prime field can be represented as a restricted sumset of the form $A\widehat{+}A=\{a+a^{\prime }:a,a^{\prime }\in A,a\ne a^{\prime }\}$. We prove that a proper multiplicative subgroup $H\le {\mathbb{F}}_p^{\ast }$ cannot satisfy $H=A\widehat{+}A$ whenever $|H|\ge 7$, and that this threshold is sharp. In fact, such a decomposition exists precisely when $|H|\in \{1,3,6\}$, and we classify all decompositions in these exceptional cases. This gives a sharp, complete resolution of the restricted-sumset analogue of the generalized Sárközy conjecture over prime fields. This significantly extends and refines previous results of Shkredov and Yip.

22 pages