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{\bf Mihail Kolountzakis}
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{\bf Lattice Tilings by Cubes: Whole, Notched and Extended}
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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 Haj\'os) and give
a new equivalent form of Haj\'os'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.

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