arXiv subjects

Yves Meyer

Publications and source records attributed to Yves Meyer.

6 recordsLinked to original sources

arXiv2026

Self-similar Delone sets and Pisot numbers

We consider Delone point patterns with self-similarity. Under mild conditions, the similarity factor is a Pisot number if and only if the pattern is uniformly discrete. The classical case is a Meyer set $\Lambda$ with $\Lambda\supset \theta\Lambda$ for some $\theta>1,$ for which $\theta$ must be a Pisot number or a Salem number. When $\Lambda$ contains several similar copies of itself, the case of a Salem number drops out for $\theta<2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.

math.DS↗

arXiv2026

Quasicrystal model sets from overlapping self-similar attractors

We give a simple computational approach to mathematical quasicrystals, combining cut-and-project methods with self-similarity. Starting with a Pisot unit $\beta$ and an iterated function system $g_k(z)=\beta z +w_k, \ k=1,...,m$ in a corresponding ring of algebraic integers, we take the attractor $A$ of the conjugate system as an acceptance window. This yields a unique cut-and-project model set $\Lambda$ in the complex plane which fulfils $\Lambda=\bigcup g_k(\Lambda) .$ We describe an algorithm which directly determines $\Lambda ,$ avoiding the difficulties with the fractal structure of $A.$ Classical constructions are based on tiles $A$ of different shape. The present study continues work on models with overlaps, as introduced by Gummelt (1996), Baake and Grimm (2013), Hejda and Pelantov\'a (2016), and Hare, Mas\'akov\'a, and V\'avra (2018). In our approach, the overlaps provide a natural decoration of $\Lambda .$ The method is illustrated with a variety of pentagonal examples.

math.MG↗

arXiv2025

Asymptotic properties of zeros of Riemann zeta function

We try to define the sequence of zeros of the Riemann zeta function by an intrinsic property. Let $(z_k)_{k\in \mathbb{N}}$ be the sequence of nontrivial zeros of $\zeta(s)$ with positive imaginary part. We write $z_k= 1/2+i\tau_k$ (RH says that these $\tau_k$ are all real). Then the sequence $(\tau_k)_{k\in \mathbb{N}},$ satisfies the following asymptotic relation \[\sum_{k\in\mathbb{N}}\frac{2x}{x^2+\tau_k^2}\simeq \frac12\log\frac{x}{2\pi}+\sum_{n=1}^\infty \frac{a_n}{x^n},\,\,x\to +\infty\] where $a_{2n+1}=2^{-2n-2}(8-E_{2n})$, $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n).$ Are there other sequences $(\alpha_k)_{k\in \mathbb{N}},$ of real or complex numbers enjoying this property? These problems are addressed in this note.

math.NT↗

arXiv2024

Properties of BV-G structures + textures decomposition models. Application to road detection in satellite images

In this paper we present some theoretical results about a structures-textures image decomposition model which was proposed by the second author. We prove a theorem which gives the behavior of this model in different cases. Finally, as a consequence of the theorem we derive an algorithm for the detection of long and thin objects applied to a road networks detection application in aerial or satellite images.

math.FA↗

arXiv2014

Quasicrystals, model sets, and automatic sequences

We survey mathematical properties of quasicrystals, first from the point of view of harmonic analysis, then from the point of view of morphic and automatic sequences. Nous proposons un tour d'horizon de propriétés mathématiques des quasicristaux, d'abord du point de vue de l'analyse harmonique, ensuite du point de vue des suites morphiques et automatiques.

math-ph↗

arXiv2013

Stable sampling and Fourier multipliers

We study the relationship between stable sampling sequences for bandlimited functions in $L^p(\R^n)$ and the Fourier multipliers in $L^p$. In the case that the sequence is a lattice and the spectrum is a fundamental domain for the lattice the connection is complete. In the case of irregular sequences there is still a partial relationship.

math.CA↗