The structure of sets with cube-avoiding sumsets

TL;DR

This paper characterizes the structure of dense subsets in finite abelian groups whose sumsets avoid certain large subsets, showing they are nearly contained in lower-dimensional structured sets.

Contribution

It proves that sets avoiding a specific subset in their sumset are essentially contained in bounded-dimension structured sets, extending understanding of sumset avoidance in finite abelian groups.

Findings

Sets avoiding Z_0^n in their sumset are nearly contained in lower-dimensional structured sets.

The structure of such sets is independent of the ambient dimension n.

The result applies to dense subsets in finite abelian groups with specific avoidance properties.

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 $E_1+\dots+E_d$ avoids $Z_0^n$ then $E_1, \dots, E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1' \times G^{I^c}, \dots, E_d' \times G^{I^c}$, where the size of $I \subset [n]$ is non-zero and independent of $n$, and $E_1',\dots,E_d'$ are subsets of $G^{I}$ such that the sumset $E_1'+\dots+E_d'$ avoids $Z_0^I$.