Fuglede's conjecture, the one dimensional case

Abstract: Fuglede conjectured that a bounded measurable set (in $\mathbb{R}^n$) is spectral if and only if it is a tile. The conjecture was also confirmed by Fuglede for sets whose tiling complement is lattice and for spectral sets one of whose spectrums is a lattice. The conjecture was disproved by Tao by constructing a spectral set in $\mathbb{Z}_3^5$, which is not a tile and lifted it to the $5$ dimensional Euclidean space.

The conjecture is open only in dimensions 1 and 2. The 1 dimensional case is directly connected with the one of finite cyclic groups and to the so called Coven-Meyerowitz conjecture. One of the main aims of the talk is to present some of the methods developed that lead to our recent results.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)