Digit sets of integral self-affine tiles with prime determinant

Jian-Lin Li¹

¹ College of Mathematics and Information Science Shaanxi Normal University Xi'an 710062, P.R. China

Studia Mathematica, Tome 177 (2006) no. 2, pp. 183-194

Résumé

Let $M\in M_{n}(\mathbb{Z})$ be expanding such that $|\!\det(M)|=p$ is a prime and $p\mathbb{Z}^n\not\subseteq M^{2}(\mathbb{Z}^n)$. Let $D\subset\mathbb{Z}^n$ be a finite set with $|D|=|\!\det(M)|$. Suppose the attractor $T(M,D)$ of the iterated function system $\{\phi_{d}(x)=M^{-1}(x+d)\}_{d\in D}$ has positive Lebesgue measure. We prove that (i) if $D\not\subseteq M(\mathbb{Z}^n)$, then $D$ is a complete set of coset representatives of $\mathbb{Z}^n/M(\mathbb{Z}^n)$; (ii) if $D\subseteq M(\mathbb{Z}^n)$, then there exists a positive integer $\gamma$ such that $D=M^{\gamma}D_{0}$, where $D_{0}$ is a complete set of coset representatives of $\mathbb{Z}^n/M(\mathbb{Z}^n)$. This improves the corresponding results of Kenyon, Lagarias and Wang. We then give several remarks and examples to illustrate some problems on digit sets.

Détail

DOI : 10.4064/sm177-2-7

Keywords: mathbb expanding det prime mathbb subseteq mathbb subset mathbb finite set det suppose attractor iterated function system phi has positive lebesgue measure prove subseteq mathbb complete set coset representatives mathbb mathbb subseteq mathbb there exists positive integer gamma gamma where complete set coset representatives mathbb mathbb improves corresponding results kenyon lagarias wang several remarks examples illustrate problems digit sets