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Jan Mazáč

Publications and source records attributed to Jan Mazáč.

17 recordsLinked to original sources
arXiv2026

On the long-range order of the Spectre tilings

The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first Čech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics.

math.DS↗
arXiv2026

Frieze patterns and aperiodic tilings of the plane

This short note provides two examples of aperiodic frieze patterns of the plane, supported on the rhombic Penrose tiling and the Godrèche--Lançon--Billard tiling. That is, we provide a decoration of their vertices with positive integers which satisfy the diamond rule, in analogy to the usual (in)finite frieze patterns as defined by Conway and Coxeter.

math.CO↗
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↗
arXiv2026

On almost periodicity in crystalline measures

Meyer defined crystalline measures as tempered distributions $μ$ such that both $μ$ and its Fourier transform $\widehatμ$ are pure-point Radon measures of locally finite support. He conjectured that every crystalline measure is almost periodic as a tempered distribution. Favorov constructed a counterexample and asked whether crystalline measures are at least almost periodic as general distributions. To resolve Favorov's question, we first show that the almost periodicity of a crystalline measure is characterised in terms of its translation boundedness, in any class of Radon measures, tempered distributions, or general distributions. We then construct a crystalline Fourier eigenmeasure that fails to be translation bounded even as a distribution. We finally construct a crystalline measure that fails to be a~Fourier quasicrystal (in particular, it fails to be slowly increasing), but it is an almost periodic tempered distribution whose Fourier transform is even a norm almost periodic measure. Our examples fully resolve the questions of Meyer and Favorov and sharply delineate the class boundary of translation boundedness. They also demonstrate the unusual behaviour of crystalline measures beyond the class of Fourier quasicrystals.

math.FA↗
arXiv2026

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↗
arXiv2025

Diffraction of plane waves, spherical waves, and beyond

We review the diffraction theory for plane waves and establish its connection to the diffraction of Besicovitch almost periodic functions, extending the theory to an unbounded setting and providing explicit formulas. Then, we give an alternative proof that the diffraction of a spherical wave in $\mathbb{R}^d$ is a single sphere, which was recently shown in \cite{BKM25}. After developing a suitable framework for working with spherically symmetric measures, including a radial analogue of the usual Lebesgue decomposition, we introduce the notion of radial almost periodicity. In particular, we define a space of Besicovitch radially almost periodic functions and show that this space contains precisely the functions whose radial part is Besicovitch almost periodic. The paper concludes with a diffraction analysis of these functions.

math.FA↗
arXiv2025

On the absolute value of the autocorrelations of the Thue-Morse sequence

Recently, Baake and Coons proved several results on the average size of the autocorrelations of the Thue--Morse sequence. They also considered the absolute value of the autocorrelations, and showed that the average value of the autocorrelations is zero. In particular, they showed that $\sum_{n\leqslant x}|η(n)|=o(x^α)$ for any $α>\log(3)/\log(4)$. In this paper, we sharpen this result, providing upper and lower bounds for $α$. On the way to our lower bounds, we obtain the structure of the linear representation of the point-wise product of two $k$-regular sequences, which may be of independent interest.

math.NT↗
arXiv2025

Diffraction of the Hat and Spectre tilings and some of their relatives

The diffraction spectra of the Hat and Spectre monotile tilings, which are known to be pure point, are derived and computed explicitly. This is done via model set representatives of self-similar members in the topological conjugacy classes of the Hat and the Spectre tiling, which are the CAP and the CASPr tiling, respectively. This is followed by suitable reprojections of the model sets to represent the original Hat and Spectre tilings, which also allows to calculate their Fourier--Bohr coefficients explicitly. Since the windows of the underlying model sets have fractal boundaries, these coefficients need to be computed via an exact renormalisation cocycle in internal space.

math.MG↗
arXiv2025

Exact renormalisation for patch frequencies in inflation systems

This note provides an explicit way of calculating the patch frequencies in geometric realisations of primitive substitutions using exact renormalisation relations. Further, we profit from these results to obtain the patch frequencies in the symbolic case as well as in other suspensions (under mild assumptions on the substitution). We illustrate this procedure on the Fibonacci example.

math.DS↗
arXiv2024

A naturally appearing family of Cantorvals

The aim of this note is to show the existence of a large family of Cantorvals arising in the projection description of primitive two-letter substitutions. This provides a common and naturally occurring class of Cantorvals.

math.DS↗
arXiv2024

Measures, modular forms, and summation formulas of Poisson type

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call $k$-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct $k$-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn-Gonçalves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional $k$-spherical measures.

math.NT↗
arXiv2024

On the Fibonacci tiling and its modern ramifications

In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well-known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher-dimensional phenomena and results. This should give some entry points to the modern literature on the subject.

math.MG↗
arXiv2023

Patch frequencies in Penrose rhombic tilings

This short exposition presents an algorithm for an exact calculation of patch frequencies for the rhombic Penrose tiling. We recall a construction of Penrose tilings via dualisation, and by extending the known method for obtaining vertex configurations, we obtain the desired algorithm. It is then used to determine the frequencies of several particular large patches which appear in the literature. The analogous approach is also explained for the Ammann-Beenker tiling.

math.DS↗
arXiv2023

Correlation functions of the Rudin-Shapiro sequence

In this paper, we show that all odd-point correlation functions of the balanced Rudin--Shapiro sequence vanish and that all even-point correlation functions depend only on a single number, which holds for any weighted correlation function as well. For the four-point correlation functions, we provide a more detailed exposition which reveals some arithmetic structures and symmetries. In particular, we show that one can obtain the autocorrelation coefficients of its topological factor with maximal pure point spectrum among them.

math.DS↗
arXiv2022

Fibonacci direct product variation tilings

The direct product of two Fibonacci tilings can be described as a genuine stone inflation rule with four prototiles. This rule admits various modifications, which lead to 48 different inflation rules, known as the direct product variations. They all result in tilings that are measure-theoretically isomorphic by the Halmos--von Neumann theorem. They can be described as cut and project sets with characteristic windows in a two-dimensional Euclidean internal space. Here, we analyse and classify them further, in particular with respect to topological conjugacy.

math.DS↗
arXiv2020

On generalized self-similarities of cut-and-project sets

Cut-and-project sets $Σ\subset\mathbb{R}^n$ represent one of the types of uniformly discrete relatively dense sets. They arise by projection of a section of a higher-dimensional lattice to a suitably oriented subspace. Cut-and-project sets find application in solid state physics as mathematical models of atomic positions in quasicrystals, the description of their symmetries is therefore of high importance. We focus on the question when a linear map $A$ on $\mathbb{R}^n$ is a self-similarity of a cut-and-project set $Σ$, i.e.\ satisfies $AΣ\subsetΣ$. We characterize such mappings $A$ and provide a construction of a suitable cut-and-project set $Σ$. We determine minimal dimension of a lattice which permits construction of such a set $Σ$.

math-ph↗