A Weyl-type theorem for Diophantine approximations driven by LCA groups and applications

TL;DR

This paper extends classical Diophantine approximation results by proving a Weyl-type equidistribution theorem for actions of LCA groups on tori, with applications to harmonic analysis and ergodic theory.

Contribution

It introduces a Weyl-type equidistribution theorem for LCA group actions, providing new foundational results in harmonic analysis and ergodic theory for these groups.

Findings

Every LCA group action admits a decomposition into uniquely ergodic subsystems.

Established Bohr orthogonality of characters along Folner sequences.

Proved a Wiener-type theorem characterizing the discrete part of measures.

Abstract

We investigate actions of locally compact Abelian (LCA) groups on the torus $\mathbb{T}^n$, motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger equidistribution theorem of Weyl type: every such action admits a decomposition into uniquely ergodic subsystems. The proof of this result is based on a characterization of unique ergodicity for actions of amenable groups on compact metric spaces. As consequences, we establish several foundational results for LCA groups, including the Bohr orthogonality of characters along arbitrary Folner sequences, a Bohr mean formula for almost periodic functions, and a Wiener-type theorem on LCA groups characterizing the discrete part of a Borel probability measure through its Fourier transform.