2025/08/11 by Sigrid Grepstad, Grepstad, Sigrid, Mihail N. Kolountzakis +3
Computer Science · Materials Science · Mathematics · #11H16 #52B20 #52C22 #Advanced Combinatorial Mathematics #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2508.07972
openalex publication_date 2025/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < vol(M). Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set F (we can even take F to be a finite union of polytopes) such that F+L is a tiling and F+M is a packing. If we do not require measurability of F it is often possible that a set F can be found tiling with both L and M even when L and M have different volumes, for instance if their intersection is trivial. We also show here that such a set can never be bounded if L and M have different volumes.