Translate of horospheres and counting problems

Mohammadi, Amir; Golsefidy, Alireza Salehi

doi:10.1353/ajm.2014.0037

Cited by 8 publications

(34 citation statements)

References 31 publications

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“…In the rank one case (n = 2) it was shown by Eskin and McMullen [5] that the quantity in Equation 4 is asymptotic to the volume of B(x, R) (times a suitable constant). The analogous result for higher rank (n > 2) was established by Mohammadi and Salehi-Golsefidy [13]. Our first theorem is an effective form of this result for G = SL n (R).…”

Section: Introductionsupporting

confidence: 75%

“…A horospherical subgroup is minimal if it is the unipotent radical of a maximal parabolic subgroup, and it is maximal if it is the unipotent radical of a minimal parabolic subgroup. 2 In [13] this theorem is proved in much greater generality, e.g. G does not necessarily have to be Q-split.…”

Section: Introductionmentioning

confidence: 99%

“…Another situation in which we know how to control simultaneous expansion and contraction was the subject of the recent work of Mohammadi and Salehi-Golsefidy [13]. They provide necessary and sufficient conditions under which translates of certain homogeneous measures (including the maximal horospherical measures) converge, and they determine the limiting measures in the cases of convergence.…”

Section: Introductionmentioning

confidence: 99%

“…It is our objective in this paper to establish the rates of convergence for the main results in [13] in the space of unimodular lattices. Our first theorem establishes effective equidistribution for translates of maximal horospherical measures.…”

Section: Introductionmentioning

confidence: 99%

“…To prove the above theorem we only need to use the effective equidistribution for directions coming from the interior of A. Consequently, our proof of Theorem 5 can be adapted to prove [13,Theorem 3] using only the wavefront lemma of Eskin-McMullen [5].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Effective equidistribution of translates of maximal horospherical measures in the space of lattices

Dabbs¹,

Kelly²,

Li³

2016

JMD

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Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient conditions under which certain translates of homogeneous measures converge, and they determined the limiting measures in the cases of convergence. The class of measures they considered includes the maximal horospherical measures. In this paper we prove the corresponding effective equidistribution results in the space of unimodular lattices. We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. Then we address the problem of determining the error terms in two counting problems also considered by Mohammadi and Salehi-Golsefidy. In the first problem, we determine an error term for counting the number of lifts of a closed horosphere from an irreducible, finite-volume quotient of the space of positive definite n × n matrices of determinant one that intersect a ball with large radius. In the second problem, we determine a logarithmic error term for the Manin conjecture of a flag variety over Q.

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Section: Introductionsupporting

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Effective equidistribution of translates of maximal horospherical measures in the space of lattices

Dabbs¹,

Kelly²,

Li³

2016

JMD

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Count lifts of non-maximal closed horocycles on $$SL_N(\mathbb {Z}) \backslash SL_N(\mathbb {R})/SO_N({\mathbb {R}})$$

Zhang

2022

Monatsh Math

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Expanding measures: Random walks and rigidity on homogeneous spaces

Prohaska

Sert

Shi

2023

Forum of Mathematics, Sigma

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Let G be a real Lie group, $\Lambda <G$ a lattice and $H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures $\mu $ on H and, applying recent work of Eskin–Lindenstrauss, prove that $\mu $ -stationary probability measures on $G/\Lambda $ are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on $G/\Lambda $ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\Lambda $ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting.

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Translate of horospheres and counting problems

Cited by 8 publications

References 31 publications

Effective equidistribution of translates of maximal horospherical measures in the space of lattices

Effective equidistribution of translates of maximal horospherical measures in the space of lattices

Count lifts of non-maximal closed horocycles on $$SL_N(\mathbb {Z}) \backslash SL_N(\mathbb {R})/SO_N({\mathbb {R}})$$

Expanding measures: Random walks and rigidity on homogeneous spaces

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