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Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the spectral-to-tiling direction for those whose prime factors grow rapidly. Until now no cyclic group with an arbitrary number of distinct divisors was known to satisfy it.
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A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.
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