Rank of matrices with entries from a multiplicative group

TL;DR

This paper derives lower bounds on the rank of matrices with entries from small-rank multiplicative groups and applies these bounds to properties of finite point sets and sumsets in Euclidean space.

Contribution

It introduces new lower bounds on matrix rank based on multiplicative group structure and applies these to geometric and additive combinatorics problems.

Findings

Distance sets in Euclidean space generate high-rank multiplicative groups.

Small-rank multiplicative groups cannot contain large sumsets.

New bounds connect matrix rank with additive and multiplicative structure.

Abstract

We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets.