The structure of sets with cube-avoiding sumsets

Abstract

We prove that if $d\ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots ,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\cdots +E_d$ avoids $Z_0^n$ then $E_1,\dots ,E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1^{\prime }\times G^{I^c},\dots ,E_d^{\prime }\times G^{I^c}$, where the size of $I\subset [n]$ is non-zero and independent of $n$, and $E_1^{\prime },\dots ,E_d^{\prime }$ are subsets of $G^I$ such that the sumset $E_1^{\prime }+\cdots +E_d^{\prime }$ avoids $Z_0^I$.

12 pages