A quantitative version of the non-abelian idempotent theorem

TL;DR

This paper provides a quantitative extension of the non-abelian idempotent theorem, showing that subsets with bounded algebra norm can be approximated by sums of a limited number of subgroup cosets, with bounds expressed as a triply tower in the norm.

Contribution

It introduces a quantitative framework for the non-abelian idempotent theorem, establishing explicit bounds on the decomposition of subsets with small algebra norm.

Findings

1. Subsets with algebra norm at most M can be expressed as sums of a bounded number of subgroup cosets.

2. The bound L is a triply tower function in M, providing explicit quantitative control.

3. The result extends the classical non-abelian idempotent theorem with precise quantitative bounds.

Abstract

Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative version of the non-abelian idempotent theorem.