Fuglede's conjecture for a union of two intervals

TL;DR

This paper proves that for unions of two intervals in the real line, being a spectral set is equivalent to tiling the real line through translations, confirming Fuglede's conjecture in this case.

Contribution

It establishes the equivalence between spectrality and tiling for unions of two intervals, a specific case of Fuglede's conjecture.

Findings

Spectral sets among unions of two intervals are exactly those that tile the real line.

The paper confirms Fuglede's conjecture for this particular class of sets.

Provides a complete characterization of spectral unions of two intervals.

Abstract

We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations.