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 (pi) be a sequence of positive numbers with P = sum pi < oo. For a real number x in [0, P], let (ci) and (di) be two sequences defined as follows: c1 = 1 if p1 \leq x, c1 = 0 otherwise; if c1, ..., ci-1 are already defined, let ci = 1 if c1 p1+...+ci-1 pi-1 \leq x-pi, ci = 0 otherwise; d1 = 0 if sumj > 1 pj \geq x, d1 = 1 otherwise; if d1, ..., di-1 are already defined, let di = 0 if sumj > i pj \geq x- sumj < i pj, di = 1 otherwise.
If sum ci pi = x (sum di pi = x), then sum ci pi (sum di pi) is called the greedy (lazy) expansion of x. More generally, sum ai pi is an expansion of x if ai in {0, 1} for every i and if sum ai pi = x.
The authors investigate these expansions in case pi = 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

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