Entropy and diffraction of the $k$-free points in $n$-dimensional lattices

TL;DR

This paper investigates the entropy and diffraction patterns of $k$-free points in $n$-dimensional lattices, revealing their complex structure and spectral properties despite having unbounded holes.

Contribution

It explicitly calculates the entropy and diffraction spectra of $k$-free points in lattices, providing new insights into their structure and spectral characteristics.

Findings

Calculated entropy of $k$-free points in lattices.

Determined diffraction spectra for these sets.

Analyzed the impact of unbounded holes on structure.

Abstract

We consider the $k$th-power-free points in $n$-dimensional lattices and explicitly calculate their entropies and diffraction spectra. This is of particular interest since these sets have holes of unbounded inradius.