# Digit sets of integral self-affine tiles with prime determinant

Studia Mathematica (2006)

• Volume: 177, Issue: 2, page 183-194
• ISSN: 0039-3223

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## Abstract

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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 ${{\varphi }_{d}\left(x\right)={M}^{-1}\left(x+d\right)}_{d\in D}$ 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 $D={M}^{\gamma }D₀$, 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.

## How to cite

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Jian-Lin Li. "Digit sets of integral self-affine tiles with prime determinant." Studia Mathematica 177.2 (2006): 183-194. <http://eudml.org/doc/284806>.

@article{Jian2006,
abstract = {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 $\{ϕ_\{d\}(x) = M^\{-1\}(x+d)\}_\{d∈ D\}$ 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 $D = M^\{γ\}D₀$, 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.},
author = {Jian-Lin Li},
journal = {Studia Mathematica},
keywords = {iterated function systems},
language = {eng},
number = {2},
pages = {183-194},
title = {Digit sets of integral self-affine tiles with prime determinant},
url = {http://eudml.org/doc/284806},
volume = {177},
year = {2006},
}

TY - JOUR
AU - Jian-Lin Li
TI - Digit sets of integral self-affine tiles with prime determinant
JO - Studia Mathematica
PY - 2006
VL - 177
IS - 2
SP - 183
EP - 194
AB - 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 ${ϕ_{d}(x) = M^{-1}(x+d)}_{d∈ D}$ 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 $D = M^{γ}D₀$, 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.
LA - eng
KW - iterated function systems
UR - http://eudml.org/doc/284806
ER -

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