# Riesz Bases of Exponentials on Unbounded Multi-tiles

## TL;DR

This paper establishes the existence of Riesz bases of exponentials for certain unbounded sets in R^d that satisfy multi-tiling and admissibility conditions, extending previous results from bounded domains.

## Contribution

It proves the existence of Riesz bases for unbounded multi-tiles, broadening the scope from bounded domains and including submulti-tiles and frames.

## Key findings

- Riesz bases exist for unbounded multi-tiles satisfying admissibility.
- Results extend known bounded domain cases to unbounded sets.
- Includes cases of submulti-tiles and frames of exponentials.

## Abstract

We prove the existence of Riesz bases of exponentials of L^2(Omega), provided that Omega in R^d is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case.