34,979 research outputs found
Folner tilings for actions of amenable groups
We show that every probability-measure-preserving action of a countable
amenable group G can be tiled, modulo a null set, using finitely many finite
subsets of G ("shapes") with prescribed approximate invariance so that the
collection of tiling centers for each shape is Borel. This is a dynamical
version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable
groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we
prove that, for every countably infinite amenable group G, the crossed product
of a generic free minimal action of G on the Cantor set is Z-stable.Comment: Minor revisions. Final versio
A generalised Euler-Poincaré formula for associahedra
We derive a formula for the number of flip-equivalence classes of tilings of an n-gon by collections of tiles of shape dictated by an integer partition λ. The proof uses the Euler–Poincaré formula; and the formula itself generalises the Euler–Poincaré formula for associahedr
Tilings by -Crosses and Perfect Codes
The existence question for tiling of the -dimensional Euclidian space by
crosses is well known. A few existence and nonexistence results are known in
the literature. Of special interest are tilings of the Euclidian space by
crosses with arms of length one, known also as Lee spheres with radius one.
Such a tiling forms a perfect code. In this paper crosses with arms of length
half are considered. These crosses are scaled by two to form a discrete shape.
We prove that an integer tiling for such a shape exists if and only if
or , . A strong connection of these tilings to binary
and ternary perfect codes in the Hamming scheme is shown
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