# 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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topJian-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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