An Unusual Continued Fraction

Badziahin, D.; Shallit, J.

Abstract

We consider the real number σ with continued fraction expansion [a0, a1, a2,...] = [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16,...], where ai is the largest power of 2 dividing i + 1. We show that the irrationality measure of σ2 is at least 8/3. We also show that certain partial quotients of σ2 grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.

Citation

Badziahin, D., & Shallit, J. (2016). An Unusual Continued Fraction. Proceedings of the American Mathematical Society, 144(5), 1887-1896. https://doi.org/10.1090/proc/12848

• Journal Article Type: Article
• Acceptance Date: May 24, 2015
• Online Publication Date: Sep 15, 2015
• Publication Date: May 1, 2016
• Deposit Date: Jun 23, 2015
• Publicly Available Date: Jun 26, 2015
• Journal: Proceedings of the American Mathematical Society
• Print ISSN: 0002-9939
• Electronic ISSN: 1088-6826
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 144
• Issue: 5
• Pages: 1887-1896
• DOI: https://doi.org/10.1090/proc/12848
• Public URL: https://durham-repository.worktribe.com/output/1435522
• Related Public URLs: http://arxiv.org/abs/1505.00667