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A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups

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Abstract

Fuglede’s conjecture states that a subset \(\Omega \subseteq \mathbb {R}^{n}\) with positive and finite Lebesgue measure is a spectral set if and only if it tiles \(\mathbb {R}^{n}\) by translation. However, this conjecture does not hold in both directions for \(\mathbb {R}^n\), \(n\ge 3\). While the conjecture remains unsolved in \(\mathbb {R}\) and \(\mathbb {R}^2\), cyclic groups are instrumental in its study within \(\mathbb {R}\). This paper introduces a new tool to study spectral sets in cyclic groups and, in particular, proves that Fuglede’s conjecture holds in \(\mathbb {Z}_{p^{n}qr}\).

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Acknowledgements

The author expresses his gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of the presentation of this paper.

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  1. Zhejiang Lab, Hangzhou, 311100, China

    Tao Zhang

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Zhang, T. A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups. Combinatorica 44, 393–416 (2024). https://doi.org/10.1007/s00493-023-00076-x

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