Flaherty and Wang studied Haar-type multiwavelets and multi-tiles. The information on what digit sets give multi-attractors with positive Lebesgue measure is very limited. In this note, we give a few classes of digit sets leading to multiattractors with positive measure. The attractors we obtain include the Haar-type multi-tiles of Flaherty and Wang. | Turk J Math 25 (2001) , 535 – 543. ¨ ITAK ˙ c TUB On the Lebesgue Measure of Self-Affine Sets ˙ Ibrahim Kırat Abstract Flaherty and Wang studied Haar-type multiwavelets and multi-tiles. The information on what digit sets give multi-attractors with positive Lebesgue measure is very limited. In this note, we give a few classes of digit sets leading to multiattractors with positive measure. The attractors we obtain include the Haar-type multi-tiles of Flaherty and Wang. Key Words: Multi-attractors, self-affine tiles, iterated function systems. 1. Introduction Unless otherwise stated, we assume that B (or Bi ) is an expanding integral matrix in Mn (Z), ., all its eigenvalues λi have modulus > 1. Let |det B| = q and let D ⊆ Zn be a set of q distinct vectors, called a q-digit set. The affine maps wj defined by wj (x) = B −1 (x + dj ), dj ∈ D, 1 ≤ j ≤ q, are all contractions under a suitable norm in Rn (see [8, pp. 29-30]). The family {wj }qj=1 is called an iterated function system (IFS) and there is a unique non-empty compact set S satisfying T = qj=1 wj (T ) ([4], [7]). T is called the attractor of the system and is given explicitly by T := T (B, D) = { ∞ X B −i dji : dji ∈ D}. i=1 We use µ(T ) to denote the Lebesgue measure of the set T . We call T an integral selfaffine tile if it has a positive measure. In general, the classification of all digit sets D 2000 Mathematics Subject Classification. Primary 52C22 535 KIRAT with µ(T ) > 0 is a complicated problem. The case that µ(T ) = 1 is of particular interest and we call such T a Haar tile. Gr¨ ochenig and Madych [6] showed that χT generates a compactly supported orthonormal wavelet basis of L2 (Rn ) if and only if µ(T ) = 1. We now consider generalized iterated function systems [2], [3], [5]. Following the notation and terminology of Flaherty and Wang [5], let Cn denote the the space of all non-empty compact subsets of Rn . Let || · || be a norm on Rn . We define the Hausdorff metric on Cn with respect
