Asymptotics of the k‐free diffraction measure via discretisation

Nick Rome,Efthymios Sofos

doi:10.1112/blms.12450

Abstract

We determine the diffraction intensity of the $k$-free integers near the origin.

Highlights

Point sets in Euclidean space exhibiting pure point diffraction play an important role in the theory of aperiodic order as mathematical models of quasicrystals
Baake and Coons [1] studied the fluctuation of the density of this set by considering the scaling behaviour of the diffraction measure νk, given by Zk(ε) = νk((0, ε])/νk({0}), as ε → 0+
We prove that a power law holds for k-free integers, confirming the conjectured behavior: Theorem 1.1

Summary

Introduction

Point sets in Euclidean space exhibiting pure point diffraction play an important role in the theory of aperiodic order as mathematical models of quasicrystals. Baake and Coons [1] studied the fluctuation of the density of this set by considering the scaling behaviour of the diffraction measure νk, given by Zk(ε) = νk((0, ε])/νk({0}), as ε → 0+. They used a sieving argument to show lim log Zk(ε) = 2 − 1/k. We shall see that the Riemann hypothesis implies a much stronger approximation of the diffraction intensity by a power law; we are not aware of a previous connection between the Riemann hypothesis and aperiodic structures.

The discretisation approach

Discretisation

Zero-free region information