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arXiv · 2607.26534

Fuglede's Conjecture on Cyclic Groups of Square-Free Order: The Case of Rapidly Growing Prime Factors

Gábor Somlai
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Abstract

The main result of the paper is that an inductive argument is established to prove the Fuglede's conjecture for an infinite sequence of square-free order cyclic groups. The tile-to-spectral direction of the conjecture holds for all square-free cyclic groups, whereas we prove the spectral-tiling direction only for those groups among them whose prime factors grow rapidly. To achieve this, we develop an inductive technique with roots in a previous paper by the author co-authored with Fallon, Kiss, and Mayeli. Following recent preprints, Fuglede's conjecture remains open only for finite cyclic groups, and until now, there was no cyclic group for which the conjecture was known that possessed an arbitrary number of distinct divisors.

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Gábor Somlai. 2026-07-29. Fuglede's Conjecture on Cyclic Groups of Square-Free Order: The Case of Rapidly Growing Prime Factors. https://arxiv.org/abs/2607.26534

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