Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that , where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results of Kenyon, Lagarias and Wang. We then give several remarks and examples to illustrate some problems on digit sets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-2-7,
author = {Jian-Lin Li},
title = {Digit sets of integral self-affine tiles with prime determinant},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {183-194},
zbl = {1123.28009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-2-7}
}
Jian-Lin Li. Digit sets of integral self-affine tiles with prime determinant. Studia Mathematica, Tome 173 (2006) pp. 183-194. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-2-7/