Tiling and weak tiling in [Formula: see text]

Gergely Kiss; Dávid Matolcsi; Máté Matolcsi; Gábor Somlai

doi:10.1007/s43670-023-00073-7

Tiling and weak tiling in ${(Z_{p})}^{d}$

Sampl Theory Signal Process Data Anal. 2024;22(1):1. doi: 10.1007/s43670-023-00073-7. Epub 2023 Dec 4.

Authors

Gergely Kiss^{1

2}, Dávid Matolcsi², Máté Matolcsi^{1

3}, Gábor Somlai^{1

2}

Affiliations

¹ Alfréd Rényi Institute of Mathematics, HUN-REN, Reáltanoda u. 13-15, 1053 Budapest, Hungary.
² Eötvös Loránd University, Pázmány Péter sétány 1/C, 1111 Budapest, Hungary.
³ Department of Analysis, Institute of Mathematics, Budapest University of Technology and Economics (BME), Müegyetem rkp. 3, 1111 Budapest, Hungary.

Abstract

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups ${(Z_{p})}^{d}$ , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d = 1, 2$ , and prove some partial results for $d = 3$ .

Keywords: Abelian p-groups; Fuglede’s conjecture; Spectral sets; Translational tiles; Weak tiling.