Absence of absolutely continuous diffraction spectrum for certain S-adic tilings

TL;DR

This paper investigates the diffraction spectra of S-adic tilings, establishing conditions under which the absolutely continuous component is absent, thereby advancing understanding of quasicrystal models.

Contribution

It provides a sufficient condition for the absence of absolutely continuous diffraction spectrum in S-adic tilings and proves this for most binary block substitutions.

Findings

Sufficient condition for zero absolutely continuous component

Proved absence of absolutely continuous spectrum for most binary S-adic tilings

Enhances understanding of diffraction properties in quasicrystal models

Abstract

Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for "almost all" binary block-substitution cases and thus prove the absence of the absolutely continuous diffraction spectrum for most of the S-adic tilings from a family of binary block substitutions.