The Minimal Decay of Matrix Coefficients for Classical Groups

Li, Jian-Shu

doi:10.1007/978-94-011-0141-7_8

Cited by 24 publications

(21 citation statements)

References 18 publications

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“…The value of the supremum has been estimated or computed in [HT,L95,LZ96,O98,O02,N03]. The estimate (3.9) can be considered as a quantitative version of property (T ).…”

Section: Definition 32 a Group G Has Kazhdan Property (T ) If One Omentioning

confidence: 99%

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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“…The value of the supremum has been estimated or computed in [HT,L95,LZ96,O98,O02,N03]. The estimate (3.9) can be considered as a quantitative version of property (T ).…”

Section: Definition 32 a Group G Has Kazhdan Property (T ) If One Omentioning

confidence: 99%

Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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“…(v) X = G/Γ with G as in (iii) and Γ any lattice commensurable with an irreducible congruence lattice. Verification of these claims depends on the fact that matrix coefficients of nontrivial irreducible representations π of almost simple simply connected groups are in L p for some p = p(π) (see [BW,Co1,Ho,HM,Li,LZ,Oh]). This implies that…”

Section: ) Regularity Under Perturbation and Geometric Comparison Amentioning

confidence: 99%

Counting lattice points

Gorodnik

Nevo

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

104

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Abstract. For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence.The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality.A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ .We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties.We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points.

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“…Let p(G) be the smallest real number such that for any p G G, p is strongly L q for any q > p(G) (cf. [7]). Similarly let pi^(G) be the smallest real number such that for any p e GK, the K-mainx coefficients of p are L^G^-integrable for any q > PK^G).…”

Section: \(P{h)v^w)\^q(h)\\v\\-\\w\\mentioning

confidence: 99%

“…The number p(G) in other classical group cases was calculated by Li [7] and was given an upper bound by Li and Zhu [8] in exceptional split group cases. We remark that the numbers r(G) in Corollary C coincide with p(G) calculated in [7] for all classical real split groups except Bn type.…”

Section: Thenp(g) < R(g) Andpk(g) < R(g)mentioning

confidence: 99%

Tempered subgroups and representations with minimal decay of matrix coefficients

Oh¹

1998

Bul. Soc. Math. France

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The Minimal Decay of Matrix Coefficients for Classical Groups

Cited by 24 publications

References 18 publications

Quantitative ergodic theorems and their number-theoretic applications

Quantitative ergodic theorems and their number-theoretic applications

Counting lattice points

Tempered subgroups and representations with minimal decay of matrix coefficients

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