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• W2963159605 abstract "Abstract This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group G , acting on suitable homogeneous spaces G / H . The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G /Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} (Γ <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∖</m:mo> </m:math> setminus G ). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples." @default.
• W2963159605 created "2019-07-30" @default.
• W2963159605 creator A5045347861 @default.
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• W2963159605 date "2016-05-12" @default.
• W2963159605 modified "2023-09-23" @default.
• W2963159605 title "Best possible rates of distribution of dense lattice orbits in homogeneous spaces" @default.
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• W2963159605 doi "https://doi.org/10.1515/crelle-2016-0001" @default.
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