Spectral sets and weak tiling

Mihail N. Kolountzakis, Nir Lev, Máté Matolcsi

Research output: Contribution to journalArticlepeer-review


A set Ω ⊂ Rd is said to be spectral if the space L2(Ω) admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that Ω is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in Rd . The proof was based on a new geometric necessary condition for spectrality, called “weak tiling”. In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure.

Original languageEnglish
Article number31
JournalSampling Theory, Signal Processing, and Data Analysis
Issue number2
StatePublished - Dec 2023


  • Equidecomposability
  • Fuglede’s conjecture
  • Polytopes
  • Spectral set
  • Tiling

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Signal Processing
  • Radiology Nuclear Medicine and imaging
  • Computational Mathematics


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