Self-Similar Lattice Tilings
- Cite this article as:
- Grochenig, K. & Haas, A. J Fourier Anal Appl (1994) 1: 131. doi:10.1007/s00041-001-4007-6
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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.