scholarly journals Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

2020 ◽  
Vol 373 (5) ◽  
pp. 3357-3374
Author(s):  
Mahbub Alam ◽  
Anish Ghosh
Keyword(s):  
Diophantine Approximation ◽  
Homogeneous Spaces

scholarly journals On orbits of unipotent flows on homogeneous spaces, II

1986 ◽  
Vol 6 (2) ◽  
pp. 167-182 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Invariant Subspace ◽  
Lebesgue Measure ◽  
Compact Subset ◽  
Lie Group ◽  
Diophantine Approximation ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Semisimple Lie Group ◽  
Unipotent Subgroup ◽  
Unipotent Flows

AbstractWe show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

scholarly journals Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds

1998 ◽  
Vol 148 (1) ◽  
pp. 339 ◽  
Author(s):  
D. Y. Kleinbock ◽  
G. A. Margulis
Keyword(s):  
Diophantine Approximation ◽  
Homogeneous Spaces

scholarly journals Best possible rates of distribution of dense lattice orbits in homogeneous spaces

2018 ◽  
Vol 2018 (745) ◽  
pp. 155-188 ◽  
Author(s):  
Anish Ghosh ◽  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Upper Bound ◽  
Homogeneous Spaces ◽  
Automorphic Representation ◽  
Upper And Lower Bounds ◽  
Duality Principle ◽  
Wide Range ◽  
Dense Orbits

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 L^{2} (Γ \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.

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