Developments in non-integer bases. (English) Zbl 0906.11008

Erdős, Pál; Komornik, V.

Acta Math. Hung. 79, No. 1-2, 57-83 (1998).

For a fixed real number \(q>1\), the sequence \(0=y_0<y_1<y_2<\dots \) of those real numbers \(y\) is considered which have at least one representation of the form \[ y=\varepsilon_0+\varepsilon_1 q+\dots +\varepsilon_n q^n, \quad n\geq 0,\;\varepsilon_i\in \{0,1\}. \] It is shown that \(y_{k+1}-y_k\) tends to zero for all \(q\) between 1 and \(2^{1/4}\), except possibly the square root of the second Pisot number \(\sqrt{p_2}\approx 1.175\). Moreover, for each such \(q\) there exists a sequence \((\varepsilon_i)\), \(\varepsilon_i\in \{0,1\}\), such that \(\sum_{i>1} \varepsilon_i q^{-i}=1\). In particular, for each such \(q\) there even exists a development containing all possible finite variations of the digits 0 and 1.

Reviewer: Peter Kirschenhofer (Leoben)

MSC:

11A67	Other number representations
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Keywords:

digital expansions; non-integer bases