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An unusual continued fraction


Authors: Dzmitry Badziahin and Jeffrey Shallit
Journal: Proc. Amer. Math. Soc. 144 (2016), 1887-1896
MSC (2010): Primary 11J70, 11J82; Secondary 11Y65, 11A55
DOI: https://doi.org/10.1090/proc/12848
Published electronically: September 15, 2015
MathSciNet review: 3460151
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Abstract: We consider the real number $ \sigma $ with continued fraction expansion $ [a_0, a_1, a_2,\ldots ] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots ]$, where $ a_i$ is the largest power of $ 2$ dividing $ i+1$. We show that the irrationality measure of $ \sigma ^2$ is at least $ 8/3$. We also show that certain partial quotients of $ \sigma ^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.


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Additional Information

Dzmitry Badziahin
Affiliation: Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham, DH1 3LE, United Kingdom
Email: dzmitry.badziahin@durham.ac.uk

Jeffrey Shallit
Affiliation: School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: shallit@cs.uwaterloo.ca

DOI: https://doi.org/10.1090/proc/12848
Received by editor(s): May 4, 2015
Received by editor(s) in revised form: May 26, 2015
Published electronically: September 15, 2015
Additional Notes: The research of the first author was supported by EPSRC Grant EP/L005204/1.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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