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arXiv · 1411.2419

Beta-expansions of rational numbers in quadratic Pisot bases

Tomáš Hejda·Wolfgang Steiner
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Abstract

We study rational numbers with purely periodic Rényi $β$-expansions. For bases $β$ satisfying $β^2=aβ+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $γ(β)=1$, i.e., that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic $β$-expansion. A simple algorithm for determining the value of $γ(β)$ for all quadratic Pisot numbers $β$ is described.

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Tomáš Hejda, Wolfgang Steiner. 2014-11-10. Beta-expansions of rational numbers in quadratic Pisot bases. https://doi.org/10.4064/aa8260-11-2017

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