Diophantine Approximation and Automorphic Spectrum

Abstract

This paper establishes quantitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and in a number of important cases are shown to be best possible. In general, it is shown that the rate of diophantine approximation is determined by the spectrum of the automorphic representation, and thus subject to the generalized Ramanujan conjectures.