GGT seminar: Andritsch

Speaker: Konstantin Andritsch

Title: Klein Sails and Geometric Continued Fractions

Abstract: The classical Gauss--Kuzmin distribution describes the asymptotic distribution of continued fraction digits. Geometrically, this may be interpreted as the equidistribution of faces of so called Klein sails in R².
In this talk we are going to revisit the geometric construction of the continued fraction expansion in arbitrary dimension. We present the strategy of the proof of the three-dimensional analogue of the Gauss--Kuzmin distribution and discuss some of its consequences. The tools used in the proof are based on results in homogeneous dynamics and are motivated by works of Kontsevich and Suhov. More precisely, the key object of study is a cross-section for the diagonal flow on SL₃(R)/SL₃(Z) which is intimately connected to the geometry of three dimensional Klein sails. The principal technical contribution is the proof of finiteness of the associated cross-sectional measure, a result that enables the derivation of the limiting face statistics.