The paper is devoted to the normal tiling whose tiles are uniformly bounded and general connected closed sets instead of being restricted to polytopes or convex sets. We estimate the number of neighbors of a tile in the normal tiling and develop various novel techniques to derive lower and upper bounds. The bounds for lattice tilings are shown to be optimal.

On the Number of Neighbors in Normal Tiling

A cluster is the union of a finite number of cubes from the standard partition ofn-dimensional Euclidean space into unit cubes. If there is lattice tiling by translates of a cluster, then must there be a lattice tiling by translates of the cluster in which the translation vectors have only integer coordinates? In this article we prove that if the interior of the cluster is connected and the dimension is at most three, then the answer is affirmative.

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Sándor Szabó

Papers

A reduction of lattice tiling by translates of a cubical cluster