# John Blackman (Durham) - A geometric algorithm for multiplying continued fractions with applications to the p-adic Littlewood conjecture

Dates: | 1 October 2019 |
---|---|

Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Speaker: | John Blackman |

Join us for this research seminar, part of the Number Theory Seminar Series.

Abstract : In recent years, the problem of deducing the continued fraction expansion of px from the continued fraction expansion of x has become a topic of increased interest. This is due in part to a reformulation of the p-adic Littlewood conjecture – an open problem in Diophantine approximation – which pertains to the behaviour of the partial quotients of xp^k as k tends to infinity (for p a fixed prime).

In this talk, we will discuss how one can interpret multiplication of a continued fraction by some integer n as a map between the Farey complex and the 1/n-scaled Farey complex. In turn, this allows us to interpret integer multiplication of continued fractions as a replacement of one triangulation on an orbifold with another triangulation. Using this geometric setting, we will then construct a reformulation of the p-adic Littlewood Conjecture and discuss how one can improve on known bounds for this problem.

### Speaker

John Blackman

Organisation: Durham University