On the transcendence of real numbers with a regular expansion.
type: Item
instance of: scholarly article
title: On the transcendence of real numbers with a regular expansion.
zbMATH Open document ID: 1052.11052
DOI: 10.1016/S0022-314X(03)00054-4
author: Boris Adamczewski
author: Julien Cassaigne
published in: Journal of Number Theory
publication date: 2003-12-03T00:00:00Z
review text: This paper is concerned with the long-standing problem: ``How regular or random is the \(b\)-ary expansion of an algebraic irrational number?'' where \(b\) is any integer greater than one. It is well known that the \(b\)-ary expansion of a rational number is ultimately periodic. It has been conjectured that the \(b\)-ary expansion of an irrational number is totally random, and several results show that if such expansion is ``too regular'', then the number is transcendental. Transcendence results have been obtained from a recent purely combinatorial condition formulated by Ferenczi and Mauduit. In particular, this condition has been used to obtain the transcendence of real numbers whose \(b\)-ary expansion is a Sturmian sequence. Here, the authors generalize this result by using the Ferenczi-Mauduit condition to prove the transcendence of real numbers whose \(b\)-ary expansion is the coding of an irrational rotation on the circle with respect to a partition in two intervals or arises from a non-periodic three-interval exchange transformation.
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 2012761
zbMATH Keywords: transcendence
zbMATH Keywords: coding of rotation
zbMATH Keywords: three-interval exchange
reviewed by: F. Blanchet-Sadri
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2012761