Additive dimension and the growth of sets

TL;DR

This paper introduces the additive dimension concept, linking it to sumset growth, and demonstrates rapid growth of sumsets for small multiplicative subgroups, with applications to sum-product problems and decompositions.

Contribution

It develops the theory of additive dimension and applies it to analyze sumset growth, providing new insights into sum-product phenomena and subgroup decompositions.

Findings

Additive dimension is closely connected to sumset growth.

Small multiplicative subgroups have rapidly growing sumsets.

Applications include sum-product estimates and decomposition results.

Abstract

We develop the theory of the additive dimension ${\rm dim} (A)$, i.e. the size of a maximal dissociated subset of a set $A$. It was shown that the additive dimension is closely connected with the growth of higher sumsets $nA$ of our set $A$. We apply this approach to demonstrate that for any small multiplicative subgroup $\Gamma$ the sequence $|n\Gamma|$ grows very fast. Also, we obtain a series of applications to the sum--product phenomenon and to the Balog--Wooley decomposition--type results.