Tuesday 26 April 2022 at 15:00

Organised in collaboration with the School of Mathematics, University of Bristol, UK

Rational Points on Horospheres, Random Abelian Cayley Graphs, and the Frobenius Coin Problem

Andreas Strömbergsson, Uppsala University, Sweden

I will discuss a result about asymptotic equidistribution of certain point sets in the homogeneous space SL(d,Z)\SL(d,R), i.e. the space of lattices in the Euclidean space R^d. This result was first proved by Einsiedler, Mozes, Shah and Shapira (2016). As I will explain, the result has applications to limit laws for various metric parameters of random Cayley graphs of a large finite abelian group, and also to the so called Frobenius coin problem. I will discuss a new proof of the equidistribution result, which leads to an explicit rate of convergence. This proof makes use of tools from algebraic geometry (recent work of Erdelyi and Toth giving bounds on matrix Kloosterman sums) and tools from geometry of numbers (an integration formula by Rogers from 1955).

Joint work with Daniel El-Baz and Min Lee.

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