Convergence of Fourier--Bohr coefficients for regular Euclidean model sets

TL;DR

This paper provides an elementary proof for the convergence of Fourier--Bohr coefficients in regular Euclidean model sets, avoiding abstract machinery and using standard exponential sum estimates and Poisson summation.

Contribution

It introduces a new elementary proof for the convergence of Fourier--Bohr coefficients in regular Euclidean model sets, simplifying previous approaches.

Findings

Elementary proof of Fourier--Bohr coefficient convergence

Applicability to regular Euclidean model sets with regular windows

Avoidance of abstract dynamical systems and harmonic analysis machinery

Abstract

It is well known that the Fourier--Bohr coefficients of regular model sets exist and are uniformly converging, volume-averaged exponential sums. Several proofs for this statement are known, all of which use fairly abstract machinery. For instance, there is one proof that uses dynamical systems theory and another one based on Meyer's theory of harmonious sets. Nevertheless, since the coefficients can be defined in an elementary way, it would be nice to have an alternative proof by similarly elementary means, which is to say by standard estimates of exponential sums under an appropriate use of the Poisson summation formula. Here, we present such a proof for the class of regular Euclidean model sets, that is, model sets with Euclidean physical and internal spaces and topologically regular windows with almost no boundary.