# Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions

## TL;DR

This paper develops a renormalisation approach to analyze pair correlations and their Fourier transforms in substitution tilings, providing criteria for spectral types and illustrating with examples including the Frank-Robinson tiling.

## Contribution

It introduces a geometric inflation rule method to derive exact renormalisation relations for pair correlation measures in self-similar and self-affine tilings, extending understanding beyond Pisot substitutions.

## Key findings

- Criteria for absence of absolutely continuous spectrum
- Application to block substitution examples
- Analysis of the Frank-Robinson tiling with singular continuous spectrum

## Abstract

For point sets and tilings that can be constructed with the projection method, one has a good understanding of the correlation structure, and also of the corresponding spectra, both in the dynamical and in the diffraction sense. For systems defined by substitution or inflation rules, the situation is less favourable, in particular beyond the much-studied class of Pisot substitutions. In this contribution, the geometric inflation rule is employed to access the pair correlation measures of self-similar and self-affine inflation tilings and their Fourier transforms by means of exact renormalisation relations. In particular, we look into sufficient criteria for the absence of absolutely continuous spectral contributions, and illustrate this with examples from the class of block substitutions. We also discuss the Frank-Robinson tiling, as a planar example with infinite local complexity and singular continuous spectrum.