Ergodic theory and the duality principle on homogeneous spaces

Gorodnik, Alexander; Nevo, Amos

doi:10.1007/s00039-014-0257-8

Cited by 11 publications

(13 citation statements)

References 48 publications

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“…After stating the result, we will explain a general strategy for how to analyze the distribution of orbits on infinite volume homogeneous spaces. Our exposition is based on [GW07,GN]. We also refer to [L99,N02,Ma02,LP03,G03,G04,GM05,Gu10,N10,MW11] for related results.…”

Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

“…where We remark that the formulation of Theorem 9.2 is optimal in the generality in which it is stated; see [GN,§1.6.1]. However, if we additionally assume that the stabilizer of a point x in X is semisimple, then the stated results can be considerably improved.…”

Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

“…First let us recall that it has been shown in [GW07] that for continuous functions φ with compact support, the asymptotic formula (9.6) holds for all x ∈ X whose Γ-orbit in X is dense. Second, it possible to formulate a suitable quantitative ergodic theorem for all functions of bounded support which are in the Lebesgue spaces L p , as follows (see [GN,Theorem 1.5] …”

Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

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Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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Abstract. We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

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Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

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Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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“…For larger groups the situation is different. Examples of actions of large groups satisfying Í analogues for compactly supported, integrable functions are given in Theorem 1.1 of [15]. In this context (example 5.1 in §5) we show that certain infinite Z 2 actions satisfy the Í analogue for all integrable functions.…”

mentioning

confidence: 80%

Symmetric Birkhoff sums in infinite ergodic theory

Aaronson¹,

Kosloff²,

Weiss³

2016

Ergod. Th. Dynam. Sys.

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“…Our approach is to reduce the study of ergodic properties of lattice orbits on the homogeneous space G/H, via the quantitative duality principle developed in [17], to the ergodic properties of the orbits of the stability group H in the space Γ \ G.…”

Section: 7mentioning

confidence: 99%

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

Ghosh

Gorodnik

Nevo

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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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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Ergodic theory and the duality principle on homogeneous spaces

Cited by 11 publications

References 48 publications

Quantitative ergodic theorems and their number-theoretic applications

Quantitative ergodic theorems and their number-theoretic applications

Symmetric Birkhoff sums in infinite ergodic theory

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

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