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arXiv · 2512.23079

An exceptional set of uniformly spread Kakutani tilings of the line

Yotam Smilansky
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Abstract

The {\alpha}-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - {\alpha}, for a fixed {\alpha} in (0,1). A simple inflation-substitution procedure produces tilings of the real line and their associated Delone sets. We show that there are precisely five distinct values of min({\alpha}, 1 - {\alpha}) for which these sets are uniformly spread, meaning that they are a bounded displacement of a lattice. The proof of this surprising fact combines the construction and analysis of a related family of primitive substitution tilings, Solomon's criterion for uniform spreadness, and a classification of Pisot-Vijayaraghavan polynomials.

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Yotam Smilansky. 2025-12-28. An exceptional set of uniformly spread Kakutani tilings of the line. https://arxiv.org/abs/2512.23079

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