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Descriptions of the Characteristic Sequence of an Irrational

Published online by Cambridge University Press:  20 November 2018

Tom C. Brown
Tom C. Brown*
Affiliation:
Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia V5A 1S6
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Abstract

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Let α be a positive irrational real number. (Without loss of generality assume 0 < α < 1.) The characteristic sequence of α is

f(α) =f1f2 ···, where fn = [(n + 1)α] - [nα].

We make some observations on the various descriptions of the characteristic sequence of α which have appeared in the literature. We then refine one of these descriptions in order to obtain a very simple derivation of an arithmetic expression for [nα] which appears in A. S. Fraenkel, J. Levitt, and M. Shimshoni [17]. Some concluding remarks give conditions on n which are equivalent to fn = 1.

Keywords

10L10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 1 , 01 March 1993 , pp. 15 - 21
Copyright
Copyright © Canadian Mathematical Society 1993

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