An unusual continued fraction

Badziahin, Dzmitry; Shallit, Jeffrey

doi:10.1090/proc/12848

An unusual continued fraction
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by Dzmitry Badziahin and Jeffrey Shallit

Proc. Amer. Math. Soc. 144 (2016), 1887-1896

DOI: https://doi.org/10.1090/proc/12848

Published electronically: September 15, 2015

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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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