arXiv subjects

Yasushi Nagai

Publications and source records attributed to Yasushi Nagai.

9 recordsLinked to original sources

arXiv2025

Overlapping substitutions and tilings

We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector encodes the patch frequency of the resulting tiling. The patch frequencies are shown to be uniformly convergent, implying that the corresponding dynamical system is uniquely ergodic. Under mild assumptions, we further prove that the associated expansion constant is always an algebraic integer. In general, overlapping substitutions may yield a patch with illegal (partial) overlaps of tiles, even if it is locally consistent. We provide a sufficient condition for an overlapping substitution to be consistent, ensuring that no such illegal tiles emerge. Finally, we construct many intriguing one-dimensional overlapping substitutions and present higher dimensional examples from Delone multi-sets with inflation symmetry.

math.CO↗

arXiv2023

Monochromatic arithmetic progressions in automatic sequences with group structure

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue--Morse and Rudin--Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence $\left\{d_n\right\}$ of differences along which the maximum length $A(d_n)$ of a monochromatic arithmetic progression (with fixed difference $d_n$) grows at least polynomially in $d_n$. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution.

math.CO↗

arXiv2021

Absence of absolutely continuous diffraction spectrum for certain S-adic tilings

Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for "almost all" binary block-substitution cases and thus prove the absence of the absolutely continuous diffraction spectrum for most of the S-adic tilings from a family of binary block substitutions.

math.DS↗

arXiv2021

On arithmetic progressions in non-periodic self-affine tilings

We study the repetition of patches in self-affine tilings in R^d. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence/non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in R^d.

math.DS↗

arXiv2021

On long arithmetic progressions in binary Morse-like words

We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.

math.CO↗

arXiv2018

A General Framework for tilings, Delone sets, functions and measures, and their interrelation

We define a general framework that includes objects such as tilings, Delone sets, functions and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in the literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object P to one that is MLD with P. Moreover we show that, for an object P and a class S of objects, a mild condition on them assures that there exists some Q in S that is MLD with P. As an application, we study pattern equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information of the original object up to MLD in a quite general setting.

math.MG↗

arXiv2018

The Common Structure For Objects In Aperiodic Order And The Theory Of Local Matching Topology

In aperiodic order, non-periodic but "ordered" objects such as tilings, Delone sets, functions and measures are investigated. In this article we depict the common structure of these objects by using the general framework of abstract pattern spaces. In particular, using the common structure we define local matching topology and uniform structure for objects such as tilings in quite a general space and a symmetry group. We prove Hausdorff property of the topology and the completeness of the uniform structure under a mild assumption. We also prove finite local complexity implies the compactness of the continuous hull and often the converse holds.

math.MG↗

arXiv2018

A new relation between geometric and dynamical properties of objects such as tilings and Delone sets

Let P be an object such as tiling, Delone set and weighted Dirac comb. There corresponds a dynamical system to P, called the corresponding dynamical system. Such dynamical systems are geometric analogues of symbolic dynamics. It is well-known that there are correspondences between geometric properties of P and properties of the corresponding dynamical system. In this article we give a new correspondence. In other words, we characterize the property that the group of topological eigenvalues for the corresponding dynamical system is not discrete, in terms of a geometric property of P.

math.DS↗

arXiv2015

Distribution of Patches in Tilings and Spectral Properties of Corresponding Dynamical Systems

A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes the nature of the tiling. In the literature, properties of this dynamical system were investigated by studying how patches distribute in each tiling. In this article we conversely research distribution of patches from properties of the corresponding dynamical systems. We show periodic structures are hidden in tilings which are not necessarily periodic. Our results throw light on inverse problem of deducing information of tilings from information of diffraction measures, in a quite general setting.

math.DS↗