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arXiv · 2507.19604

On finite $\beta$-expansions for the set of natural numbers

Túlio O. Carvalho·Catharina M. Moreira
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Abstract

We present a study of the problem of finiteness of the $\beta$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $\beta$-expansion of 1 is not a non-decreasing sequence. We show a class of simple $\beta$-numbers which display a puzzling behaviour, in the sense that an infinite subset of such $\beta$ satisfy $F_1$, whereas a complementary infinite subset does not. This puzzle is organized by the residue class modulo 3 of an integer coefficient of the main family of polynomials discussed. We prove that, when $F_1$ does not hold, the complement of ${{\rm Fin}(\beta)}$ is infinite. Finally, we give a concise sufficient condition for $F_1$, which is the finitude of the $\beta$-expansion of $(\beta-\lfloor \beta\rfloor)^2$.

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BibTeXRIS

Túlio O. Carvalho, Catharina M. Moreira. 2025-07-25. On finite $\beta$-expansions for the set of natural numbers. https://arxiv.org/abs/2507.19604

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