Squiral diffraction

TL;DR

This paper investigates a planar substitution system related to the squiral inflation rule, demonstrating that it exhibits purely singular continuous diffraction with an explicitly computable Riesz product measure.

Contribution

It introduces a new planar substitution system with singular continuous diffraction, extending the understanding of diffraction in higher dimensions.

Findings

Diffraction is purely singular continuous for balanced weights.

The diffraction measure is a two-dimensional Riesz product.

Explicit calculation of the diffraction measure is achieved.

Abstract

The Thue-Morse system is a paradigm of singular continuous diffraction in one dimension. Here, we consider a planar system, constructed by a bijective block substitution rule, which is locally equivalent to the squiral inflation rule. For balanced weights, its diffraction is purely singular continuous. The diffraction measure is a two-dimensional Riesz product that can be calculated explicitly.