On the dimension of additive sets

TL;DR

This paper investigates various notions of dimension for additive sets, establishing bounds between them and demonstrating their asymptotic sharpness through the construction of large dissociated subsets.

Contribution

It improves existing inequalities relating different dimensions of additive sets and proves these bounds are asymptotically optimal using dissociated subsets.

Findings

Bounds for ratios between different dimensions are established.

Improved inequality of Lev and Yuster is presented.

Existence of large dissociated subsets is demonstrated.

Abstract

We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between these dimensions by improving an inequality of Lev and Yuster, and we show that these bounds are asymptotically sharp, using in particular the existence of large dissociated subsets of $\{0,1\}^n\subset \mathbb{Z}^n$.