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Self-Similar Lattice Tilings

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Abstract

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.

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  1. Department of Mathematics U-9, University of Connecticut, Storrs, Connecticut 06269-3009, USA

    Karlheinz Grochenig & Andrew Haas

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  1. Karlheinz Grochenig
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  2. Andrew Haas
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Grochenig, K., Haas, A. Self-Similar Lattice Tilings. J Fourier Anal Appl 1, 131–170 (1994). https://doi.org/10.1007/s00041-001-4007-6

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  • DOI: https://doi.org/10.1007/s00041-001-4007-6

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