Ergodic properties of visible lattice points

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 $\mathcal{B}$-free numbers with Sarnak's conjecture on the `randomness' of the Möbius 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.

arXiv comment26 pages, 5 figures; results presented at the Mini-Workshop "Dynamical versus Diffraction Spectra in the Theory of Quasicrystals" at the MFO in Oberwolfach in December 2014; revised version with minor corrections

References in corpus (6)

• Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
• Characterizations of model sets by dynamical systems
• Entropy and diffraction of the $k$-free points in $n$-dimensional lattices
• Dynamical versus diffraction spectrum for structures with finite local complexity
• Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
• Dynamical properties of $k$-free lattice points

Cited by in corpus (11)

• Dynamics on the graph of the torus parametrisation
• On pattern entropy of weak model sets
• A short guide to pure point diffraction in cut-and-project sets
• Aperiodic order and spherical diffraction, I: Auto-correlation of model sets
• Dynamics of $\mathcal{B}$-free sets: a view through the window
• Number-theoretic positive entropy shifts with small centraliser and large normaliser
• Periods and factors of weak model sets
• On the logarithmic probability that a random integral ideal is $\mathcal{A}$-free
• Automorphisms of Toeplitz $\mathcal{B}$-free systems
• $\mathfrak{B}$-free integers in number fields and dynamics
• Scaling of the diffraction measure of $k$-free integers near the origin