Zaremba’s Conjecture for Geometric Sequences: An Algorithm

Abstract

Even though Zaremba’s conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba’s conjecture.

MSC:

• 11A55

ACKNOWLEDGMENT

I would like to thank Anna Theorin Johansson as well as my advisor, Claire Burrin, for many engaging discussions and helpful feedback. Moreover, I am very grateful to the reviewers for their constructive comments. This work was supported by Swiss National Science Foundation grant PR00P2_201557.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author.