On the binary expansions of algebraic numbers
type: Item
instance of: scholarly article
title: On the binary expansions of algebraic numbers
zbMATH Open document ID: 1076.11045
author: D. H. Bailey
author: Jonathan M. Borwein
author: Richard Crandall
author: Carl Pomerance
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2005-07-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/51828
full work available at URL: metadc785402
review text: Using concepts from additive number theory and results on binary evaluations and partial series, the authors prove that for a real algebraic number \(y\) of degree \(D> 1\) the number \(\#(|y|, N)\) of 1-bits in the expansion of \(|y|\) through bit position \(N\) is greater than \(CN^{1/D}\), where \(C> 0\) is a number depending only on \(y\), and \(N\) is sufficiently large. By means of this result the transcendency of a class of reals \(\sum_{n\geq 0} 2^{-f(n)}\) is obtained, where the integer-valued function \(f\) grows sufficiently fast. For example, for any real \(\alpha> 1\) the number \(\sum_{n\geq 0} 2^{-\lfloor\alpha^n\rfloor}\) is transcendental, and so the transcendency of the Kempner-Mahler number \(\sum_{n\geq 0} 2^{-2^n}\) is re-established. Further, they also study certain ``borderline'' numbers, and prove that the number \(\sum_{n\geq 0}2^{-n^2}\) is not a quadratic irrational. Moreover, some open problems are proposed.
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 2188528
zbMATH Keywords: binary-evaluation bounds
zbMATH Keywords: transcendency
zbMATH Keywords: algebraic numbers
reviewed by: Yaochen Zhu
MaRDI profile type: Publication
OpenAlex ID: W2098447376
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DOI: 10.5802/JTNB.457
description: scientific article