Tiling and weak tiling in $(\mathbb{Z}_p)^d$

TL;DR

This paper explores the relationships between tiling, weak tiling, and spectral sets in finite abelian groups, introducing a new averaging method that generalizes these concepts through a 4-tuple of functions, with characterizations for low dimensions.

Contribution

It introduces an averaging procedure in elementary p-groups that unifies tiling and spectral set concepts via a 4-tuple of functions, providing characterizations for dimensions 1 and 2, and partial results for 3.

Findings

Characterization of 4-tuples for d=1,2

Partial results for d=3

A new averaging procedure linking tiling and spectral sets

Abstract

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, 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 $d=3$.