arXiv subjects

Anna Klick

Publications and source records attributed to Anna Klick.

4 recordsLinked to original sources

arXiv2026

Renormalisation techniques for inflation systems and some of their applications

Exact renormalisation techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalisation principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for system with pure-point spectrum, the orbit separation dimension, and its relation to renormalisation. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalisation and Lyapunov exponents can be used to exclude the presence of absolutely continuous part of the spectra.

math.DS↗

arXiv2025

Pair correlations of one-dimensional model sets and monstrous covariograms of Rauzy fractals

The averaged distance structure of one-dimensional regular model sets is determined via their pair correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair correlation functions. We introduce this approach and derive two concrete examples, which display an unexpectedly complex and wild behaviour.

math.MG↗

arXiv2023

On higher dimensional arithmetic progressions in Meyer sets

In this paper we study the existence of higher dimensional arithmetic progression in Meyer sets. We show that the case when the ratios are linearly dependent over $\ZZ$ is trivial, and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $Λ$ and a fully Euclidean model set $\oplam(W)$ with the property that finitely many translates of $\oplam(W)$ cover $Λ$, we prove that we can find higher dimensional arithmetic progressions of arbitrary length with $k$ linearly independent ratios in $Λ$ if and only if $k$ is at most the rank of the $\ZZ$-module generated by $\oplam(W)$. We use this result to characterize the Meyer sets which are subsets of fully Euclidean model sets.

math.NT↗

arXiv2020

On arithmetic progressions in model sets

In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.

math.DS↗