Digit sets of integral self-affine tiles with prime determinant
Volume 177 / 2006
Abstract
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.
