Pushing the gap between tiles and spectral sets even further

Abstract: Fuglede conjectured that a bounded measurable set in a locally compact topological space endowed with Haar measure is spectral if and only if it is a tile and Fuglede also confirmed the conjecture for sets whose tiling complement is a lattice and for spectral sets one of whose spectrums is a lattice.

The conjecture was disproved by Tao in the case of finite abelian groups where the counting measure plays the role of the Haar measure. Tao constructed a spectral set in $\mathbb{Z}_3^5$ of size 6, that is not a tile. This construction was lifted to the $5$ dimensional Euclidean space, where the original conjecture was mostly studied.

Lev and Matolcsi verified Fuglede's conjecture for convex sets in $\mathbb{R}^n$ for every positive integer $n$. The key of proving the harder direction of the conjecture is to introduce the weak tiling property and prove that all spectral sets are weak tilings.

One of the goals of our work was to answer a question of Kolountzakis, Lev and Matolcsi, whether there is a weak tile that is neither a tile nor spectral. There is such a set which apparently makes it harder to prove the spectral-tile direction of the conjecture in the remaining open cases.

The other result towards structurally distinguishing spectral sets and tiles was a disproof of a conjecture of Greenfeld and Lev. They conjectured that the product of two sets is spectral if and only if both of them are spectral. A similar property holds for tiles, but the product of a non-spectral set with a spectral set can be spectral. Finally, we obtain an easy characterization of tiles using the spectral property.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)