# GAN

## Seminar

### Partial sums of excursions along random geodesics.

Seminar Room 2, Newton Institute Gatehouse

#### Abstract

In the theory of continued fractions, Diamond and Vaaler showed the following strong law: for almost every expansion, the partial sum of first n coefficients minus the largest coefficient divided by n \log n tends to a limit. We will explain how this generalizes to non-uniform lattices in SL(2, R) with cusp excursions in the quotient hyperbolic surface generalizing continued fraction coefficients. The general theorem relies on the exponential mixing of geodesic flow, in particular on the fast decay of correlations due to Ratner. Analogously, similar theorems are true for the moduli space of Riemann surfaces.#### Video

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