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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
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  • Department of Mathematics

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  • In category "Seminar"
  • In group "(Maths) Maths seminar series"
  • In group "(Maths) Number theory "
  • By Department of Mathematics

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

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Faustin Adiceam

faustin.adiceam@manchester.ac.uk

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