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ABSOLUTELY ABNORMAL AND CONTINUED FRACTION NORMAL NUMBERS

Part of: Elementary number theory Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  16 March 2016

JOSEPH VANDEHEY
JOSEPH VANDEHEY*
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA email vandehey@uga.edu
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Abstract

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In this short note, we give a proof, conditional on the generalised Riemann hypothesis, that there exist numbers $x$ which are normal with respect to the continued fraction expansion but not to any base-$b$ expansion. This partially answers a question of Bugeaud.

Keywords

continued fractionsnormal numbersArtin’s conjecture

MSC classification

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary: 11J70: Continued fractions and generalizations 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 217 - 223
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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