Pisot numbers and greedy algorithm. (English) Zbl 0919.11063

Akiyama, Shigeki

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).

Let \(\beta > 1\) and \(x > 0\). An expansion \(x = \sum_{N_0}^\infty a_i \beta^{-i}\) with integers \(0 \leq a_i < \beta\) is called a greedy expansion if \(\sum_{N+1}^\infty a_i \beta^{-i} < \beta^{-N}\) for every \(N\). The author proves that if \(\beta\) is a Pisot unit for which every \(x \in \mathbb Z[\beta]\) has a finite greedy expansion in base \(\beta\), then there is a positive constant \(c\) such that every \(x \in {\mathbb Q}\cap [0,c]\) has a purely periodic greedy expansion in base \(\beta\). This is related to a result of A. Bertrand [C. R. Acad. Sci., Paris, Sér. A 285, 419-421 (1977; Zbl 0362.10040)] and K. Schmidt [Bull. Lond. Math. Soc. 12, 269-278 (1980; Zbl 0494.10040)], which states that every \(0 < x \in \mathbb Q(\beta)\) has a periodic (but not necessary purely periodic) greedy expansion in base \(\beta\). The proof of the author’s theorem allows one to obtain an effective lower bound on \(c\) for a given \(\beta\). For example, it is shown that if \(\beta = 1.3247\dots\) is the smallest Pisot number, then \(c \geq 0.4342\) but it cannot be taken larger than \(0.6924\).
For the entire collection see [Zbl 0887.00013].

Reviewer: D.W.Boyd (Vancouver)

MSC:

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

Keywords:

Pisot number; beta expansion; purely periodic greedy expansion

Citations:

Zbl 0362.10040; Zbl 0494.10040