“Spectral implies Tiling” for three intervals revisited

Abstract

Abstract. In [2] 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.

Spectrality of a class of self-affine measures on R2 *

“…Although Fuglede's conjecture in its original form has been disproved, there is a clear connection between spectral sets and tilings, but the precise correspondence is still a mystery. On the other hand, there has been an effort to prove or disprove Fuglede's conjecture in R and R 2 , or for special classes of subsets of R n like unions of intervals [4,31] and convex bodies [27,34]. Recently, Fan et al [21] proved that Fuglede's conjecture in the p-adic field Q p holds.…”

Spectrality of generalized Sierpinski-type self-affine measures

“…However, the conjecture is still open in low dimensions n = 1, 2. As of today there has been an effort to prove or disprove Fuglede's conjecture, one can refer to [4,18,23,27] and the references therein.…”

On the Rationality of the Spectrum

“…We will restrict to dimension d = 1. In this case Fuglede's conjecture is known to hold when Ω is the union of two intervals [12] and for the case of three intervals the authors proved that Tiling implies Spectral, and except for one situation, Spectral implies tiling too, [1], [2] . In Lagarias and Wang [14] studied the structure of tiling sets T , and proved that if (Ω, T ) is a tiling pair for some Ω, then the tiling set T is periodic with an integer period and is rational, i.e.…”