DSNT Seminar: Almost optimal Diophantine exponent for SL(n)

Abstract:
We will start by describing the density of SLn(Z[1/p]) in SLn(R) in a quantitative manner along the line of work by Ghosh--Gorodnik--Nevo. The Diophantine exponent kappa for a pair of elements x,y in SLn(R) is a certain positive real number that, loosely, measures the complexity of an element gamma in SLn(Z[1/p]) such that gamma x approximates y with a prescribed error. Ghosh--Gorodnik--Nevo conjectured that \kappa should be optimal, which means kappa less than equal to 1 (after certain normalization), and proved this on certain varieties. However, for SL(n) their method gives kappa less than or equal to n-1. In this talk, we try to describe how certain automorphic techniques can improve the bound of kappa to something as good as 1+O(1/n). If time permits, we will also talk about the L^2-growth of the Eisenstein series on reductive groups. This is one of the inputs in our proof towards improved Diophantine exponent. This is a joint work with Amitay Kamber.

For more information, please contact the organiser Reza Mohammadpour.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).