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SUMMARY:Subhajit Jana (Max Planck Institute for Mathematics Bonn)
DTSTART:20220513T103000Z
DTEND:20220513T113000Z
DTSTAMP:20260423T022926Z
UID:ntsea/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ntsea/13/">A
 lmost optimal Diophantine exponent for $\\mathrm{SL}(n)$</a>\nby Subhajit 
 Jana (Max Planck Institute for Mathematics Bonn) as part of Number theory 
 by the sea\n\n\nAbstract\nWe will start by describing the density of $\\ma
 thrm{SL}_n(\\mathbb{Z}[1/p])$ in $\\mathrm{SL}_n(\\mathbb{R})$ in a quanti
 tative manner along the line of work by Ghosh--Gorodnik--Nevo. The Diophan
 tine exponent $\\kappa$ for a pair of elements $x\,y \\in\n\\mathrm{SL}_n(
 \\mathbb{R})$ 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--Go
 rodnik--Nevo\nconjectured that $\\kappa$ should be optimal\, which means $
 \\kappa \\le 1$ (after certain normalization)\, and proved this on certain
 \nvarieties. However\, for $\\mathrm{SL}(n)$ their method gives $\\kappa \
 \le n-1$. In this talk\, we try to describe how certain automorphic\ntechn
 iques can improve the bound of $\\kappa$ to something as\ngood as $1+O(1/n
 )$. If time permits\, we will also talk about the $L^2$-growth of the Eise
 nstein series on reductive groups. This is one of the inputs in our proof 
 towards improved Diophantine exponent. This is a joint work with Amitay Ka
 mber.\n
LOCATION:https://researchseminars.org/talk/ntsea/13/
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