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Beta-expansions of rational numbers in quadratic Pisot bases

Volume 183 / 2018

Tomáš Hejda, Wolfgang Steiner Acta Arithmetica 183 (2018), 35-51 MSC: Primary 11A63; Secondary 11R06, 37B10. DOI: 10.4064/aa8260-11-2017 Published online: 7 March 2018


We study rational numbers with purely periodic Rényi $\beta$-expansions. For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ to have a purely periodic $\beta$-expansion. We provide a simple algorithm for determining the infimum of $p/q\in[0,1)$ with $\gcd(q,b)=1$ and whose $\beta$-expansion is not purely periodic, which works for all quadratic Pisot numbers $\beta$.


  • Tomáš HejdaDepartment of Mathematics FNSPE
    Czech Technical University in Prague
    Trojanova 13
    12000 Praha, Czech Republic
    Department of Algebra MFF
    Charles University
    Sokolovská 49/83
    18675 Praha, Czech Republic
  • Wolfgang SteinerIRIF, CNRS UMR 8243
    Université Paris Diderot – Paris 7
    Case 7014
    75205 Paris Cedex 13, France

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