John Blackman (Durham) - A geometric algorithm for multiplying continued fractions with applications to the p-adic Littlewood conjecture
|Starts:||15:00 1 Oct 2019|
|Ends:||16:00 1 Oct 2019|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
|Who is it for:||University staff, External researchers, Adults, Alumni, Current University students|
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.
Organisation: Durham University
Travel and Contact Information
Frank Adams 1
Alan Turing Building