Abstract
We obtain a complete classification of polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings, i.e., partitions into parallel shifts of one set that is affinely similar to the initial one. In every dimension, there exist infinitely many nonequivalent polyhedral sets possessing this property. Under an additional assumption that the affine similarity is defined by an integer matrix and by integer shifts (“digits”) from different quotient classes with respect to this matrix, the only polyhedral set of this kind is a parallelepiped. Applications to multivariate wavelets and to Haar systems are discussed.
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Funding
This work was supported by the Russian Foundation for Basic Research, project nos. 19-04-01227 and 20-01-00469. The second author’s research was supported by a grant from the Government of the Russian Federation for the state support of scientific research conducted under the direction of leading researchers, project no. 14.W03.31.0031.
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Translated by I. Ruzanova
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Protasov, V.Y., Zaitseva, T.I. Self-Affine Tiling of Polyhedra. Dokl. Math. 104, 267–272 (2021). https://doi.org/10.1134/S1064562421050112
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DOI: https://doi.org/10.1134/S1064562421050112