Subhajit Jana
(Queen Mary University of London)

Mathematics Seminar

We will start by describing the density of $\mathrm{SL}_n(\mathbb{Z}[1/p])$ in $\mathrm{SL}_n(\mathbb{R})$ in a quantitative manner along the line of work by Ghosh--Gorodnik--Nevo. The Diophantine exponent $\kappa$ for a pair of elements $x,y \in \mathrm{SL}_n(\mathbb{R})$ is a certain positive real number that, loosely, measures the complexity of an element $\gamma\in\mathrm{SL}_n(\mathbb{Z}[1/p])$ such that $\gamma x$ approximates $y$ with a prescribed error. Ghosh--Gorodnik--Nevo conjectured that $\kappa$ should be optimal, which means $\kappa \le 1$ (after certain normalization), and proved this on certain varieties. However, for $\mathrm{SL}(n)$ their method gives $\kappa \le n-1$. 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.

Contact: Paul Nelson
Revised: 13.03.2024