Geodesic
Lattice tilings by cubes: Whole, notched and extended
The electronic journal of combinatorics, Tome 5 (1998)
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We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of ${\bf R}^d$ 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 rectangle has been removed from one of the corners) and show that they admit lattice tilings. This has also been 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.
DOI : 10.37236/1352
Classification : 52C22, 52C07, 42B10
Mots-clés : lattice tiling, harmonic analysis, unit cube, Fourier transform 
@article{10_37236_1352,
     author = {Mihail Kolountzakis},
     title = {Lattice tilings by cubes: {Whole,} notched and extended},
     journal = {The electronic journal of combinatorics},
     year = {1998},
     volume = {5},
     doi = {10.37236/1352},
     zbl = {0892.52017},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1352/}
}
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%A Mihail Kolountzakis
%T Lattice tilings by cubes: Whole, notched and extended
%J The electronic journal of combinatorics
%D 1998
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%U http://geodesic.mathdoc.fr/articles/10.37236/1352/
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Mihail Kolountzakis. Lattice tilings by cubes: Whole, notched and extended. The electronic journal of combinatorics, Tome 5 (1998). doi: 10.37236/1352

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