Local translations associated to spectral sets

TL;DR

This paper explores the relationship between local translation groups and spectral sets in real numbers and integers, providing insights into their structure and connections to tiling, in the context of the Fuglede conjecture.

Contribution

It establishes links between local translation groups on  and r and offers examples for sets with few elements, advancing understanding of spectral sets and tiling.

Findings

Connections between local translation groups on z and r

Examples of spectral sets with low cardinality

Relations between local translations and tilings

Abstract

In connection to the Fuglede conjecture, we study groups of local translations associated to spectral sets, i.e., measurable sets in $\br$ or $\bz$ that have an orthogonal basis of exponential functions. We investigate the connections between the groups of local translations on $\bz$ and on $\br$ and present some examples for low cardinality. We present some relations between the group of local translations and tilings.