Related papers: Tilings for Pisot beta numeration

To a given tiling a non commutative space and the corresponding C*-algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for…

In the joint work of T.Rivoal and the author, a hypergeometric construction was proposed for studing arithmetic properties of the values of Dirichlet's beta function $\beta(s)$ at even positive integers. The construction gives some bonuses…

We study Brown's definition of the probabilistic zeta function of a finite lattice as a generalization of that of a finite group. We propose a natural alternative or extension that may be better suited for non-atomistic lattices. The…

Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). We consider algebraic properties of the resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p must be…

Let $\cal T$ be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then $\cal T$ is periodic and it consists of…

A substitution $\vp$ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot $\vp$ that satisfies a no cycle condition and for…

We consider numeration systems where digits are integers and the base is an algebraic number $\beta$ such that $|\beta|>1$ and $\beta$ satisfies a polynomial where one coefficient is dominant in a certain sense. For this class of bases…

We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We…

Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…

Motivated by the study of Fibonacci-like Wang shifts, we define a numeration system for $\mathbb{Z}$ and $\mathbb{Z}^2$ based on the binary alphabet $\{0,1\}$. We introduce a set of 16 Wang tiles that admits a valid tiling of the plane…

Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d…

We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily…

Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $\delta (T)$ the number of non isomorphic indecomposable summands of $T$.…

For $b>1$ and $\alpha\beta$ a string of two digits in base $b$, let $K_1$ be the subsum of the harmonic series with only those integers having exactly one occurrence of $\alpha\beta$. We obtain a theoretical representation of such $K_1$…

Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…

Every normal periodic tiling is a strongly balanced tiling. The properties of periodic tilings by convex polygons are rearranged from the knowledge of strongly balanced tilings. From the results, we show the properties of representative…

We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1,…

The combinatorics of tilings of a hexagon of integer side-length $n$ by 120 degree - 60 degree diamonds of side-length 1 has a long history, both directly (as a problem of interest in thermodynamic models) and indirectly (through the…

Motivated by applications in reliable and secure communication, we address the problem of tiling (or partitioning) a finite constellation in $\mathbb{Z}_{2^L}^n$ by subsets, in the case that the constellation does not possess an abelian…

This article addresses the problem of enumerating the tilings of a plane by lozenges, under the restriction that these tilings be doubly periodic. Kasteleyn's Pfaffian method is applied to compute the generating function of those…