On Fuglede’s conjecture for three intervals

Abstract

In this paper, we prove the Tiling implies Spectral part of Fuglede’s cojecture for the three interval case. Then we prove the converse Spectral implies Tiling in the case of three equal intervals and also in the case where the intervals have lengths 1/2, 1/4, 1/4. Next, we consider a set Ω ⊂ R, which is a union of n intervals. If Ω is a spectral set, we prove a structure theorem for the spectrum provided the spectrum is assumed to be contained in some lattice. The method of this proof has some implications on the Spectral implies Tiling part of Fuglede’s conjecture for three intervals. In the final step in the proof, we need a symbolic computation using Mathematica. Finally with one additional assumption we can conclude that the Spectral implies Tiling holds in this case.