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SUMMARY:Optimal Diophantine Exponents and the Spectral Decomposition - Ami
 tay Kamber\, University of Cambridge
DTSTART:20220201T143000Z
DTEND:20220201T153000Z
UID:TALK169121@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION:Let q be a prime. It is simple to show that SLn(Z[1/q]) is den
 se in SLn(R )\, and we want to make this quantitative. Equivalently\, we w
 ant to study the density of the q-Hecke orbit of a point on the locally sy
 mmetric space SLn(Z[1/q]) SLn(R )/SO(n). This problem was studied in great
  generality by Ghosh-Gorodnik-Nevo\, who defined a ”Diophantine exponent
 ” to measure the density of the orbit. A similar definition appears in t
 he work of Sarnak and Parzanchevski on Golden Gates. Assuming the\nGeneral
 ized Ramanujan Conjetcure (GRC)\, we prove that the Diophantine exponents 
 in our case are optimal. Unconditionally\, we prove that the exponents are
  optimal for n=2 and n=3\, and are almost optimal for general n. The proof
  combines ”density bounds” towards GRC by Blomer with new bounds on th
 e L^2-growth of Eisenstein series in a compact domain which we develop\, a
 nd are of independent interest. Based on ongoing work with Subhajit Jana.
LOCATION:MR13
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