Kenyon Theorem Revisited

Szymon Głąb; Mateusz Kula

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arXiv · 2508.19454

Kenyon Theorem Revisited

Szymon Głąb·Mateusz Kula

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Abstract

We study sets $E(\Sigma,q)=\left\{\sum_{i=1}^\infty \sigma_iq^i\colon(\sigma_i)\in\Sigma^{\mathbb N}\right\}$ for a finite set $\Sigma\subset \mathbb R$ and $q\in(0,1)$. Under the assumption $q|\Sigma|=1$ we prove several new equivalent conditions for $E(\Sigma,q)$ to contain an interval. We give a full characterization, if additionally $|\Sigma|$ is prime.

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BibTeX RIS

Szymon Głąb, Mateusz Kula. 2025-08-26. Kenyon Theorem Revisited. https://arxiv.org/abs/2508.19454

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