Spectrality and tiling by cylindric domains

TL;DR

This paper characterizes when cylindric sets are spectral or can tile space, showing these properties depend solely on their base sets, thus linking spectrality and tiling to lower-dimensional structures.

Contribution

It provides a complete characterization of spectral and tiling properties of cylindric sets based on their base sets, extending understanding of spectral sets and tiling in higher dimensions.

Findings

A cylindric set is spectral if and only if its base is spectral.

A cylindric set can tile space by translations if and only if its base can tile.

The results connect spectrality and tiling properties of cylindric sets to those of their bases.

Abstract

A bounded set $\Omega \subset \mathbb{R}^d$ is called a spectral set if the space $L^2(\Omega)$ admits a complete orthogonal system of exponential functions. We prove that a cylindric set $\Omega$ is spectral if and only if its base is a spectral set. A similar characterization is obtained of the cylindric sets which can tile the space by translations.