2024/11/06 by Gergely Kiss, Itay Londner, Kiss, Gergely +5
Engineering · Materials Science · Mathematics · #05B45 #11B75 #20K01 (Primary) 11C08 #43A47 #51D20 #52C22 (Secondary) #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2411.03854
openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group ℤM. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions f,g, such that f(0)=g(0)=1, f∗ g=1ℤM is a functional tiling, and f, g satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.