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

Nagai, Yasushi (2021). Absence of absolutely continuous diffraction spectrum for certain S-adic tilings. Nonlinearity, 34(11) pp. 7963–7990.

DOI: https://doi.org/10.1088/1361-6544/ac2a51

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 ${\mathbb{R}}^{d}$ , 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 S-adic tilings from a family of binary block substitutions.