Integral Self-Affine Tiles in ℝ ⁿ I. Standard and Nonstandard Digit Sets

Abstract

We investigate the measure and tiling properties of integral self-affine tiles, which are sets of positive Lebesgue measure of the form T(A,@) = { £ * x A~'d^: all d } €@}, where AeM n (Z) is an expanding matrix with |det (A)| = m, and Qs ^ 2 " is a set of m integer vectors. The set Q> is called a digit set, and is called standard if it is a complete set of residues of Z"/A(Z") or arises from one by an integer affine transformation, and nonstandard otherwise. We prove that all sets T(A, Of) have integer Lebesgue measure, and study when the measure fi (T(A, Si)) ^ 0. We give a Fourier-analytic condition for /z(r(A, Si)) =t 0. We classify nonstandard digit sets in special cases, and give formulae for the measures of their associated tiles.

Self-Affine Sets and Graph-Directed Systems

“…(2) What is the Lebesgue measure for such a self-affine region? It is known that the Lebesgue measure of a self-affine tile is an integer [20]. (3) How to compute the Hausdorff dimension of T or its boundary when it is a self-affine (self-similar) region?…”

“…We call such T a self-affine tile. The geometric and algebraic properties of the self-affine tiles have been studied in great detail in literature recently (see, e.g., [1], [7], [12], [16], [18], [19], [20], [21], [30], [32]). The case for N ≥ |det(A)| arises naturally in the study of iterated function systems with overlaps [23], [28].…”

Classification of Integral Expanding Matrices and Self-Affine Tiles

“…It is easy to show that if q = 2, then T is always a tile and the D is standard. In [LW3] Lagarias and Wang proved that if q is a prime and , D)) > 0 if and only if D is a standard digit set. They also conjecture that condition (1.1) is redundant [LW3].…”

“…In [LW3] Lagarias and Wang proved that if q is a prime and , D)) > 0 if and only if D is a standard digit set. They also conjecture that condition (1.1) is redundant [LW3]. We therefore consider the self-affine tiles for the case of |det(A)| = 3 that is not covered by (1.1):…”

Topological properties of self-similar fractals with one parameter

“…In particular, if A is a similarity, then T is called a self-similar set/tile. Since the fundamental theory of self-affine tiles was established by Lagarias and Wang ( [13], [14], [15]), there have been considerable interests in the topological structure of self-affine tiles T , including but not limited to the connectedness of T ( [7], [8], [12], [1], [6]), the boundary ∂T ( [2], [19], [22]), or the interior T • of a connected tile T ( [24], [25]). Especially in R 2 , the study on the disk-likeness of T (i.e., the property of being a topological disk) has attracted a lot of attentions ( [5], [16], [23], [11], [6]).…”