What question did this study set out to answer?

To investigate the conjecture regarding the spectrality of Cartesian products of sets.

January 17, 2026

Spectrality of the product does not imply that the components are spectral

Key Points

To investigate the conjecture regarding the spectrality of Cartesian products of sets.
Constructed a specific counterexample involving two sets.
Demonstrated the existence of a tile without spectra.
Established that the Cartesian product can be spectral even if the individual sets are not.
Provided a specific example illustrating the failure of the conjecture.

Abstract

Greenfeld and Lev J. Anal. Math., 16 (2020), 409–441 conjectured that the Cartesian product of two sets A A and B B is spectral if and only if A A and B B are spectral. We construct a counterexample to this conjecture using the existence of a tile that has no spectra.

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