2018/09/10 by Liu, Bochen · 1 citation
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1809.03440
Suppose f∈ L1(ℝd), Λ⊂ℝd is a finite union of translated lattices such that f+Λ tiles with a weight. We prove that there exists a lattice L⊂ℝd such that f+L also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity. Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficient criteria to tell whether a convex polygon admits translational multi-tilings. As an application, one can easily construct symmetric (2m)-gons, for any m≥ 4, that do not multi-tile by translations. Finally, we prove a convex polygon which is not a parallelogram only admits periodic multiple tilings, if any.