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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 824.11005

**Autor: ** Erdös, Paul; Joó, István; Komornik, Vilmos

**Title: ** On the number of q-expansions. (In English)

**Source: ** Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 37, 109-118 (1994).

**Review: ** Let (p_{i}) be a sequence of positive numbers with P = **sum** p_{i} < oo. For a real number x in [0, P], let (c_{i}) and (d_{i}) be two sequences defined as follows: c_{1} = 1 if p_{1} \leq x, c_{1} = 0 otherwise; if c_{1}, ..., c_{i-1} are already defined, let c_{i} = 1 if c_{1} p_{1}+...+c_{i-1} p_{i-1} \leq x-p_{i}, c_{i} = 0 otherwise; d_{1} = 0 if **sum**_{j > 1} p_{j} \geq x, d_{1} = 1 otherwise; if d_{1}, ..., d_{i-1} are already defined, let d_{i} = 0 if **sum**_{j > i} p_{j} \geq x- **sum**_{j < i} p_{j}, d_{i} = 1 otherwise.

If **sum** c_{i} p_{i} = x (**sum** d_{i} p_{i} = x), then **sum** c_{i} p_{i} (**sum** d_{i} p_{i}) is called the greedy (lazy) expansion of x. More generally, **sum** a_{i} p_{i} is an expansion of x if a_{i} in **{**0, 1**}** for every i and if **sum** a_{i} p_{i} = x.

The authors investigate these expansions in case p_{i} = q^{-i}, where q in (1, 2) (q-expansions) and they give a new proof of the following property stated by the same authors [Bull. Soc. Math. Fr. 118, 377-390 (1990; Zbl 721.11005)]: For every 1 \leq N \leq \omega there are 2^{\omega} numbers q in (1, 2) such that 1 has exactly N different q-expansions.

**Reviewer: ** L.Tóth (Cluj)

**Classif.: ** * 11A67 Representation systems for integers and rationals

**Keywords: ** expansions of real numbers; greedy expansion

**Citations: ** Zbl 721.11005

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag