Ergodic properties of visible lattice points

TL;DR

This paper reviews the ergodic and spectral properties of visible lattice points and their relevance to dynamical systems, number theory, and diffraction, highlighting recent advances and implications for mathematical diffraction theory.

Contribution

It provides a comprehensive summary of recent results on the spectral and dynamical properties of visible lattice points and their connection to number theory and diffraction.

Findings

Spectral analysis of visible lattice points

Connections to Sarnak's conjecture and M"obius function

Implications for mathematical diffraction theory

Abstract

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $\mathscr B$-free numbers with Sarnak's conjecture on the `randomness' of the M\"obius function, another the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.