×

Rational numbers with purely periodic \(\beta \)-expansion. (English) Zbl 1211.11010

Let \(\beta>1\) be a real number and \(\xi\in[0,1)\). Following A. Rényi, [Acta Math. Acad. Sci. Hung. 8, 477–493 (1957; Zbl 0079.08901)] the \(\beta\)-expansion of \(\xi\) is defined as the sequence \(d_\beta(\xi)=(a_n)_{n\geq1}\) produced by the \(\beta\)-transformation \(T_\beta:x\mapsto \beta x \mod 1\) with \(a_n=\lfloor \beta T_{\beta}^{n-1}(\xi)\rfloor\); thus \(\xi=\sum_{n\geq 1} a_n \beta^{-n}\).
Let \(\gamma(\beta)\) denote the supremum of the real numbers \(c\in[0,1)\) such that all positive rational numbers less than \(c\) have a purely periodic \(\beta\)-expansion. The authors study \(\gamma(\beta)\) for cubic numbers \(\beta>1\). They prove that \(\gamma(\beta)>0\) holds if and only if \(\beta\) is a Pisot unit satisfying the property (F). Furthermore, if additionally \(\mathbb{Q}(\beta)\) is not totally real, then \(\gamma(\beta)\) is irrational, so in particular, \(\gamma(\beta)<1\).
As usual, a Pisot number is a real algebraic integer which is greater than \(1\) and which has all Galois conjugates (different from itself) inside the open unit disc. A Pisot unit is a Pisot number which is a unit of the ring of integers of the number field it generates. Finally, \(\beta\) is said to have property (F) if every \(x\in\mathbb Z[1/\beta] \cap [0,1)\) has a finite \(\beta\)-expansion.
This is in partial contrast to quadratic \(\beta\), where \(\gamma(\beta)\) is positive if and only if \(\beta\) is a Pisot unit satisfying (F), but in that case, \(\gamma(\beta)=1\) always holds, which follows from work of S. Akiyama [Pisot numbers and greedy algorithm, in: Győry, Kálmán (ed.) et al., Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29–August 2, 1996. Berlin: de Gruyter, 9–21 (1998; Zbl 0919.11063)], M. Hama and T. Imahashi [Comment. Math. Univ. St. Pauli 46, No. 2, 103–116 (1997; Zbl 0899.11039)] as well as K. Schmidt [Bull. Lond. Math. Soc. 12, 269–278 (1980; Zbl 0494.10040)].

MSC:

11A63 Radix representation; digital problems
11J72 Irrationality; linear independence over a field
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
28A80 Fractals
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv