Rough Approximate subgroups

TL;DR

This paper investigates the structure of rough approximate subgroups within first-order structures, establishing the existence of a type-definable normal subgroup with bounded index contained in a bounded power of the original subgroup.

Contribution

It introduces a new structural result for rough approximate subgroups, showing they contain a normal subgroup with bounded index within a bounded power of the subgroup.

Findings

Existence of a type-definable normal subgroup normalized by the approximate subgroup

Normal subgroup contained in the fourth power of the approximate subgroup

Bounded index of the subgroup in the generated group

Abstract

Given a $T$-rough definably amenable $T$-rough approximate subgroup $A$ of a group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ of bounded index in $\langle A\rangle$.