Lattice Tilings By Cubes: Whole, Notched and Extended

We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of R d by the unit cube in relation to the Minkowski Conjecture (now a theorem of Hajos) and give a new equivalent form of Hajos'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. the electronic journal of combinatorics 5 (1998), #R14 2 0. Introduction 0.1 Results. We obtai..