On Lebesgue Measure of Integral Self-Affine Sets

Ievgen V. Bondarenko; Rostyslav V. Kravchenko

doi:10.1007/s00454-010-9306-8

On Lebesgue Measure of Integral Self-Affine Sets

Ievgen V. Bondarenko, Rostyslav V. Kravchenko

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Let A be an expanding integer n×n matrix and D be a finite subset of Zⁿ. The self-affine set T=T(A,D) is the unique compact set satisfying the equality A(T)=∪_dD(T+d). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈Zⁿ, and the measure of the intersection of self-affine sets T(A,D₁)∩T(A,D₂) for different sets D₁, D₂⊂Zⁿ.

Original language	English (US)
Pages (from-to)	389-393
Number of pages	5
Journal	Discrete and Computational Geometry
Volume	46
Issue number	2
DOIs	https://doi.org/10.1007/s00454-010-9306-8
State	Published - Sep 2011

Funding

R.V. Kravchenko was partially supported by NSF grant 0503688.

Keywords

Graph-directed system
Self-affine set
Self-similar action
Tile

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-010-9306-8

Cite this

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abstract = "Let A be an expanding integer n×n matrix and D be a finite subset of Zn. The self-affine set T=T(A,D) is the unique compact set satisfying the equality A(T)=∪dD(T+d). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈Zn, and the measure of the intersection of self-affine sets T(A,D1)∩T(A,D2) for different sets D1, D2⊂Zn.",

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AU - Kravchenko, Rostyslav V.

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AB - Let A be an expanding integer n×n matrix and D be a finite subset of Zn. The self-affine set T=T(A,D) is the unique compact set satisfying the equality A(T)=∪dD(T+d). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈Zn, and the measure of the intersection of self-affine sets T(A,D1)∩T(A,D2) for different sets D1, D2⊂Zn.

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KW - Self-affine set

KW - Self-similar action

KW - Tile

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On Lebesgue Measure of Integral Self-Affine Sets

Abstract

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Keywords

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