The Fuglede conjecture for convex domains is true in all dimensions

Máté Matolcsi (Budapest University of Technology and Economics (BME), Budapest, Hungary; and Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary)

Abstract

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2 (\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\Omega \subset \mathbb{R}^d$ the “tiling implies spectral” part of the conjecture is in fact true.

To the contrary, the “spectral implies tiling” direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques.

In this paper we fully settle Fuglede’s conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set, then $\Omega$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric “weak tiling” condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.

Keywords

Fuglede’s conjecture, spectral set, tiling, convex body, convex polytope

2010 Mathematics Subject Classification

42B10, 52B11, 52C07, 52C22

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N. L. was supported by ISF Grants No. 227/17 and 1044/21 and ERC Starting Grant No. 713927.

M. M. was supported by NKFIH Grants No. K129335 and K132097.

Received 5 January 2021

Accepted 30 December 2020

Published 1 July 2022