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Lattice tilings by cubes: Whole, notched and extended. (English) Zbl 0892.52017

Electron. J. Comb. 5, Research paper R14, 11 p. (1998); printed version J. Comb. 5, 193-203 (1998).
Summary: We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of \(\mathbf {R}^d\) by the unit cube in relation to the Minkowski conjecture (now a theorem of Hajós) and give a new equivalent form of Hajós’s theorem. We also consider “notched cubes” (a cube from which a reactangle has been removed from one of the corners) and show that they admit lattice tilings. This has also been proved by S. Stein by a direct geometric method. Finally, we exhibit a new class of simple shapes that admit lattice tilings, the “extended cubes”, which are unions of two axis-aligned rectangles that share a vertex and have intersection of odd codimension. In our approach we consider the Fourier Transform of the indicator function of the tile and try to exhibit a lattice of appropriate volume in its zero-set.

MSC:

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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