On Jul 28, 2017, at 6:25 AM, James Propp <jamespropp@gmail.com> wrote:
I was thinking of translations only. So that excludes the Socolar-Taylor monotile I was thinking of, as inspiration for a counterexample in 2D (this tiling requires both rotations and reflections of the tile).
But here’s a related challenge: Construct a finite set of "lattice tiles for Z^2" (finite subsets of Z^2, not necessarily connected by adjacency), such that the maximum packing density in the lattice torus (Z/B)^2, of translates of the tiles, is strictly less than the supremum of the packing density in Z^2, again by translates, for arbitrarily large B. Certainly some such set of lattice tiles can be constructed using the Socolar-Taylor tile as a guide, but the tiles will comprise many lattice points. My challenge is therefore to find such a set that also minimizes the cardinality of lattice points (in the tile set). -Veit
More specifically, I think of a packing of Z by translates of the finite set B as being determined by a set S (a subset of the integers) such that the sets B+s are disjoint (as s ranges over S), and I call the packing periodic if the indicator function of the set S is periodic.
Sorry for not saying this sooner.
Jim