What question did this study set out to answer?

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

January 17, 2026Proceedings of the American Mathematical Society

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

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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.

Implication

Counterexample reveals that the Cartesian product being spectral does not guarantee component spectrality.

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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Cite This Study

Gábor Somlai (2026) studied this question.

synapsesocial.com/papers/696b26b2d2a12237a934a079 https://doi.org/10.1090/proc/17462

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Implication

Counterexample reveals that the Cartesian product being spectral does not guarantee component spectrality.

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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