Bounded common fundamental domains for two lattices

2025/07/01 by Sigrid Grepstad, Grepstad, Sigrid, Mihail N. Kolountzakis +1 · 2 citations
Materials Science · Mathematics · #11H16 #52B20 #52C22 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Metric Geometry (math.MG) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2507.00604

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for any two lattices L, M ⊆ ℝ^d of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set E ⊆ ℝ^d such that E tiles ℝ^d when translated by L or by M. In fact, the set E can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of E forms a Weyl--Heisenberg (Gabor) orthogonal basis of L²(ℝ^d) when translated by L and modulated by M^*, the dual lattice of M.

Citations

• Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability
• Bounded distance equivalence of cut-and-project sets and equidecomposability
• Solutions to the discrete Pompeiu problem and to the finite Steinhaus tiling problem
• Functions tiling with several lattices
• The Steinhaus tiling problem and the range of certain quadratic forms

Cited by

• Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability
• Bounded lattice tiles that pack with another lattice

Related

• Pairs of Full-Rank Lattices With Parallelepiped-Shaped Common Fundamental Domains
• Non-separable lattices, Gabor orthonormal bases and Tilings
• Bounded lattice tiles that pack with another lattice
• On common fundamental domains
• Tiling by lattices for locally compact abelian groups