NATURAL TILING, LATTICE TILING AND LEBESGUE MEASURE OF INTEGRAL SELF-AFFINE TILES

Abstract

In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice $\varGamma\subseteq\mathbb{Z}^n$ as a translation tiling set of $\mathbb{R}^n$. In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile $K$ and to determine all of the lattice tilings of $\mathbb^n$ by $K$. Moreover, we also propose and determine algorithmically another type of translation tiling of $\mathbb{R}^n$ by $K$, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set $K\cap (K+j),\ j\in\mathbb{Z}^n$, is strictly positive.