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Generalised golden ratios over integer alphabets

Baker, S. (2014) Generalised golden ratios over integer alphabets. Integers, 14. A15. ISSN 1553-1732

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It is a well known result that for β ∈ (1,1+√52) and x ∈ (0,1β−1) there exists uncountably many (ǫi)∞i=1 ∈ {0,1}N such that x = P∞i=1ǫiβ−i. When β ∈ (1+√52,2] there exists x ∈ (0,1β−1) for which there exists a unique (ǫi)∞i=1 ∈ {0,1}N such that x=P∞i=1ǫiβ−i. In this paper we consider the more general case when our sequences are elements of {0, . . . , m}N. We show that an analogue of the golden ratio exists and give an explicit formula for it.

Item Type:Article
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:46858
Uncontrolled Keywords:Beta-expansions, Dimension theory
Publisher:De Gryuter


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