Bounds in Cohen's idempotent theorem

TL;DR

This paper establishes bounds in Cohen's idempotent theorem, showing that integer-valued functions with bounded algebra norm on finite Abelian groups can be decomposed into a sum of characteristic functions of subgroup cosets with explicit bounds.

Contribution

It provides explicit bounds on the number of subgroup cosets needed to represent such functions, advancing understanding of the structure of functions with bounded algebra norm.

Findings

Decomposition of integer-valued functions into subgroup cosets

Explicit bounds on the number of cosets involved

Extension of Cohen's idempotent theorem with quantitative bounds

Abstract

We show that if $G$ is a finite Abelian group and $f$ is an integer-valued map on $G$ with algebra norm at most $M$ then there is some $L < \exp(M^{4+o(1)})$, cosets of (possibly different) subgroups $W_1,...,W_L$, and $s_1,...,s_L \in \{-1,1\}$ such that $f=\sum_i{s_i1_{W_i}}$.