The coset and stability rings

TL;DR

This paper demonstrates that subsets of discrete Abelian groups with bounded Fourier algebra norm exhibit a specific stability property, quantified by an exponential bound related to the norm.

Contribution

It establishes a new connection between Fourier algebra norms and stability properties in discrete Abelian groups, providing explicit bounds.

Findings

Sets with bounded Fourier algebra norm are exponentially stable.

Provides explicit stability bounds in terms of the Fourier algebra norm.

Links harmonic analysis with combinatorial stability concepts.

Abstract

We show that if $G$ is a discrete Abelian group and $A \subset G$ has $\|1_A\|_{B(G)} \leq M$ then $A$ is $O(\exp(\pi M))$-stable in the sense of Terry and Wolf.