Multi-tiling and Riesz bases

TL;DR

This paper extends the theory of exponential bases from tiling sets to multi-tiling sets in R^d, showing that multi-tiling sets admit Riesz bases of exponentials using Meyer's quasicrystals.

Contribution

It generalizes Fuglede's theorem by proving multi-tiling sets have Riesz bases of exponentials, broadening the class of sets with such bases.

Findings

Multi-tiling sets in R^d have Riesz bases of exponentials.

The proof utilizes Meyer's quasicrystals.

Generalizes the tiling result to multi-tiling scenarios.

Abstract

Let S be a bounded, Riemann measurable set in R^d, and L be a lattice. By a theorem of Fuglede, if S tiles R^d with translation set L, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles R^d with translation set L, S has a Riesz basis of exponentials. The proof is based on Meyer's quasicrystals.