Lattice Tilings by Cubes: Whole, Notched and Extended

We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of Rd by the unit cube in relation to the Minkowski Conjecture (now a theorem of Haj´os) and give a new equivalent form of Haj´os’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 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. 1991 Mathematics Subject Classification. Primary 52C22; Secondary 42. 

• Type: Journal
• Year: 1998
• Where: COMBINATORICS
• Authors: Mihail N. Kolountzakis
• Keywords: COMBINATORICS 1998, Cubes, Harmonic Analysis Methods, Lattice Tilings