## Pisot numbers and greedy algorithm.(English)Zbl 0919.11063

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].

### MSC:

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

### Citations:

Zbl 0362.10040; Zbl 0494.10040