"Spectral implies Tiling" for Three Intervals Revisited

TL;DR

This paper revisits the spectral implies tiling aspect of Fuglede's conjecture for three intervals, establishing new proofs and connections to algebraic geometry, with most cases confirmed and one remaining exceptional case linked to Vandermonde varieties.

Contribution

It extends the understanding of Fuglede's conjecture for three intervals by proving spectral implies tiling in most cases and relating the exceptional case to generalized Vandermonde varieties.

Findings

Spectral implies Tiling holds for two intervals due to circle intersection properties.

Most cases of three intervals satisfy spectral implies Tiling.

An exceptional case relates to generalized Vandermonde varieties.

Abstract

In \cite{BCKM} it was shown that "Tiling implies Spectral" holds for a union of three intervals and the reverse implication was studied under certain restrictive hypotheses on the associated spectrum. In this paper, we reinvestigate the "Spectral implies Tiling" part of Fuglede's conjecture for the three interval case. We first show that the "Spectral implies Tiling" for two intervals follows from the simple fact that two distinct circles have at most two points of intersections. We then attempt this for the case of three intervals and except for one situation are able to prove "Spectral implies Tiling". Finally, for the exceptional case, we show a connection to a problem of generalized Vandermonde varieties.