On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

TL;DR

This paper analyzes the spectral properties of the Thue-Morse quasicrystal's Dirac comb, revealing its complex structure through conjectures and number theory, and characterizing its spectrum up to measure zero.

Contribution

It provides a detailed spectral decomposition of the Thue-Morse quasicrystal, linking fractal sum-of-digits functions and algebraic number theory to the spectral analysis.

Findings

Spectral decomposition involves measure-zero sets and conjectures.

Asymptotic arithmetic of p-rarefied sums influences the spectrum.

Scaling of approximant measures relates to class number and regulator of quadratic fields.

Abstract

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godr\`eche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyri\`ere and M. Queff\'elec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.