Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space

Shi, Ronggang

doi:10.1090/tran/8028

Cited by 11 publications

(25 citation statements)

References 32 publications

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“…We are going to need Shi's equidistribution result ([Shi17], Corollary 1.3) for number fields. This follows from much general Theorem 1.2 of [Shi17].…”

Section: Proof Of Theorems 23 and 24mentioning

confidence: 76%

“…In this paper, among other results, we generalize the above weighted spiraling of approximates to number fields. We follow the approach of Kleinbock, Shi and Weiss and use an equidistribution result of Shi [Shi17]. We also answer a question raised by Kleinbock, Shi and Weiss regarding the equidistribution of orbits of certain flows on homogeneous spaces with respect to unbounded functions.…”

Section: Introductionmentioning

confidence: 99%

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

Alam

Ghosh

2020

Trans. Amer. Math. Soc.

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The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss [KSW17] regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M. Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove "spiraling" results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh and Tseng [AGT15, AGT14] as well as Kleinbock, Shi and Weiss [KSW17].If · is taken to be the supremum norm then c m can be taken to be 1. In [AGT15], Athreya, Ghosh and Tseng considered the problem of 'spiraling' of approximates connected to the Diophantine inequality above. LetGiven A ⊆ S m−1 , T > 0, they considered the counting functions N(x, T ) = #{(p, q) ∈ Z m × Z + : qx − p < c m |q| −1/m , 0 < q ≤ T } and N(x, T, A) = #{(p, q) ∈ Z m × Z + : qx − p < c m |q| −1/m , 0 < q ≤ T, θ(p, q) ∈ A}, and proved: A.

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“…We are going to need Shi's equidistribution result ([Shi17], Corollary 1.3) for number fields. This follows from much general Theorem 1.2 of [Shi17].…”

Section: Proof Of Theorems 23 and 24mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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

Alam

Ghosh

2020

Trans. Amer. Math. Soc.

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“…While height functions are a classical tool since the seminal work [13] by Eskin, Margulis and Mozes, the height function for our problem has to be carefully and ingeniously constructed. The techniques used in this paper are similar in spirit to those used to prove Birkhoff genericity by Chaika and Eskin in [7] and by the second author in [38]. Different methods are used instead in [28] and [39] to prove Birkhoff genericity for one parameter diagonalizable subgroup actions on homogeneous spaces: there the key step consists of proving effective double equidistribution of translates of volume measures on horospherical orbits.…”

Section: Elkies and Mcmullen Proved Inmentioning

confidence: 99%

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

Frączek¹,

Shi²,

Ulcigrai³

2018

Journal of Modern Dynamics

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In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices ASL 2 (R)/ASL 2 (Z), we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum H(1, 1) of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragović and Radnović, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Frączek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.

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“…Using the results in [BQ11], Shi showed in [Shi14] the β-contraction hypothesis for the G action on X for some β > 0. We reproduce the proof in this section with some modifications to obtain a more precise range for the exponent β.…”

Section: The Contraction Hypothesis For Sl(2r) Actionsmentioning

confidence: 99%

Bounded and divergent trajectories and expanding curves on homogeneous spaces

Khalil¹

2020

Trans. Amer. Math. Soc.

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Suppose g t is a 1-parameter Ad-diagonalizable subgroup of a Lie group G and Γ < G is a lattice. We study differentiable curves of the form ϕ : [0, 1] → U + satisfying certain non-degeneracy conditions, where U + is the expanding horospherical subgroup of g t . For a class of examples that includes products of real rank one Lie groups, we obtain sharp upper bounds on the Hausdorff dimension of the set of points s for which the forward orbit (g t ϕ(s)x 0 ) t 0 is divergent on average in G/Γ, for any basepoint x 0 ∈ G/Γ. Moreover, we prove that the set of points s for which (g t ϕ(s)x 0 ) t 0 remains bounded is winning in the sense of Schmidt. We describe applications of our results to problems in diophantine approximation. Our methods also yield the following result for square systems of linear forms: suppose ϕ(s) = sY + Z where Y ∈ GL(n, R) and Z ∈ M n,n (R). Then, the dimension of the set of points s such that ϕ(s) is singular is at most 1/2 while badly approximable points have Hausdorff dimension equal to 1.Lemma 4.5. There exists a constant C 1 > 1, such that for all x ∈ X, natural numbers n with α(n) ≥ 1/γ, t > 0 and all subintervals J ⊂ [−1, 1] of radius at least e −α(nt) , one hasProof. First, we note that for all r ∈ [−1, 1], we haveUsing positivity of f , (4.5) and a change of variable, we getThen, Fubini's theorem and the fact that ϕ is C 1+γ imply the following. J f (g (n+1)t u(ϕ(s))x) ds J 1 −1 f (g (n+1)t u(ϕ(s) + re −α(nt)φ (s) + O(e −(1+γ)α(nt) ))x) dr dsMoreover, by definition of g t and u(Y ), we haveThus, by our assumption that α(n) 1/γ, we getIn particular, by (4.7), we getSince e α(t) ∈ N, for each 1 ≤ j ≤ k, the partition P k is a refinement of P j . This implies the following inclusion.Hence, the following inequality follows from (4.8), (4.9), and the fact that f is non-negative.(4.10) Iterating (4.10), by induction, we obtain the following exponential decay integral estimate. B(M,n−1)∩J 0 f (g (m+n)t u(ϕ(s))x) ds (2C 1c ) n e −βα(nt) J∈Pm J∩B(M,n−1)∩J 0 =∅ J f (g mt u(ϕ(s))x) ds = (2C 1c ) n e −βα(nt) J 0 f (g mt u(ϕ(s))x) ds (4.11)where on the second line, we used the following consequence of P m being a partition.

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Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space

Cited by 11 publications

References 32 publications

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

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

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

Bounded and divergent trajectories and expanding curves on homogeneous spaces

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