The distribution of random continued fractions

Vaibhav Gadre (University of Glasgow)

Monday 30th October, 2023 16:00-17:00 Maths 311B

Abstract

Suppose that we use a fair coin to generate an infinite random sequence of heads and tails. We may then associate to it a continued fraction by recording the number of consecutive heads (then consecutive tails etc.) as the coefficients in a continued fraction. It is well known (as special cases of Guivarc'h–LeJan, Deroin–Kleptsyn–Navas, Blachere–Haissinski–Mathieu, G–Maher–Tiozzo) that the distribution of the continued fractions obtained from this model is singular with respect to the Lebesgue measure on the interval.  In this talk, we will survey the key ideas and further show that the distribution is strongly singular, namely that its pushforward by a quasi-symmetry remains singular. This is joint work with Aitor Azemar and answers a question posed to the second author by McMullen.

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