Continued Fractions, Interval Exchanges and Applications to Geometry
timetable:
Thu 13 Jun, 15:30 - 15:25, Aula Dini
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On the quantitative metric theory of continued fractions
speaker: Poj Lertchoosakul (PhD Student at Department of Mathematical Sciences, University of Liverpool, UK)abstract: Quantitative versions of the central results of the metric theory of continued fractions were given by C. de Vroedt. We give improvements of the bounds involved. For a real number \(\alpha\), we use the usual notation of continued fraction \(\alpha = [a_0(\alpha);a_1(\alpha),a_2(\alpha),\dotsc]\). A sample result we prove is that, given \(\epsilon>0\), \[(a_1(\alpha)\dotsm a_n(\alpha))^{\frac{1}{n}} = \prod_{k=1}^\infty \bigg(1+\frac{1}{k(k+2)}\bigg)^\frac{\log k}{\log 2} + o\Big(n^{-\frac{1}{2}}(\log n)^\frac{3}{2}(\log\log n)^{\frac{1}{2}+\epsilon}\Big)\] almost everywhere with respect to Lebesgue measure. This is a joint work with J. Hančl (Ostrava), A. Jaššová (Liverpool) and R. Nair (Liverpool).
timetable:
Thu 13 Jun, 15:30 - 15:25, Aula Dini
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