The structure of sets with cube-avoiding sumsets

Abstract

Suppose $$ is a finite abelian group, $$ is not contained in any strict coset in $$, and $$ are dense subsets of $$ such that the sumset $$ avoids $$. We show that $$ and $$ are almost entirely contained in sets defined by a bounded number of coordinates, that is, sets $$ and $$, where the size of $$ is non-zero and independent of $$, and $$ are subsets of $$ such that $$ avoids $$. Furthermore, we show that this result extends to any finite group and $$ summands for any $$.