Natural extension domains of α-odd continued fractions

Faculty Member

Florin Boca

Project Image

Continued fractions are widespread in mathematics due to their natural relationship to many fields including number theory, dynamical systems, and ergodic theory. The sequence of numerators and denominators can be generated by repeatedly applying some generating map. Changing the generating map changes the type of continued fraction expansion. We introduced the α-odd continued fractions, defined on the domain [α-2,α), which only have odd denominators.

It is often easier to study a two-dimensional version of the generating map, defined over some domain. Our paper “α-expansions of odd partial quotients” (https://arxiv.org/pdf/1806.06166.pdf) describes the domain when (-1+ √5 )/2<α<(1+ 5)/2. This project will reprove these results using an insertion and singularization algorithm, similar to the methods used by Kraaikamp for the regular α-continued fractions (see for example https://arxiv.org/pdf/1707.09321.pdf). The main tools of this project will be careful applications of matrix multiplication and limits of sequences. If time permits, we will also explore when α<(-1+5)/2 and the domain is fractal.

Team Meetings

Weekly

Project Difficulty

Intermediate

Undergrad Prerequisites

416 and 444 (or 447/424)