Diophantine Approximation and Automorphic Spectrum

Ghosh, Anish; Gorodnik, Alexander; Nevo, Amos

doi:10.1093/imrn/rns198

Cited by 31 publications

(67 citation statements)

References 36 publications

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“…Later a systematic study of this property for homogeneous varieties X was undertaken in [16], where in particular it has been shown that • every α ∈ S n is (1, 0, b)-Dirichlet in S n for any…”

Section: Proofs Of Theoremsmentioning

confidence: 99%

“…Previously Fukshansky [15] used a theorem of Hlawka [20] about approximations of real numbers by Pythagorean triples to establish Theorem 1.1 in the special case of S 1 , and showed that one can take C = 2 √ 2. In [33] a version of the above theorem was established for all n with φ 1 replaced by φ 1/2⌈log 2 (n+1)⌉ and with an explicit dependence of C on n. Later Ghosh, Gorodnik and Nevo [16] did the same with φ 1 replaced by φ 1 4 −ε for all n and with C dependent on α and ε (see §4.1 for a more precise statement of their results).…”

Section: Introductionmentioning

confidence: 97%

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Rational approximation on spheres

Kleinbock

Merrill

2015

Isr. J. Math.

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We quantify the density of rational points in the unit sphere S n , proving analogues of the classical theorems on the embedding of Q n into R n . Specifically, we prove a Dirichlet theorem stating that every point α ∈ S n is sufficiently approximable, the optimality of this approximation via the existence of badly approximable points, and a Khintchine theorem showing that the Lebesgue measure of approximable points is either zero or full depending on the convergence or divergence of a certain sum. These results complement and improve on previous results, particularly recent theorems of Ghosh, Gorodnik and Nevo.Date: May 22, 2013.

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Section: Proofs Of Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Rational approximation on spheres

Kleinbock

Merrill

2015

Isr. J. Math.

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“…, p n , q) ∈ Z n+1 lying on a quadratic variety of the form x 2 1 ± · · · ± x 2 d − y 2 = 1 to their limit points (with respect to the projective distance) lying on the unit sphere S n ⊂ R n . For general manifolds admitting group actions, Ghosh, Gorodnik and Nevo [29] also consider related Diophantine problems involving approximation by rational points. The Khintchine type theory for curves in dimensions n > 2 is simply nonexistent.…”

Section: Diophantine Approximation On Manifoldsmentioning

confidence: 99%

Rational points near manifolds and metric Diophantine approximation

Beresnevich

2012

Ann. Math.

145

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This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of R n . These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarník type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifoldsa very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions n > 2 and show that they are ubiquitous (that is uniformly distributed).

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“…Using the bounds (13) when n = 2, Ghosh, Gorodnik, and Nevo [51] showed that there exists 0 > 0, depending only on r and δ, such that for every ∈ (0, 0 ) one can find an S-integer point z ∈ X(O S ) satisfying ||x − z|| ∞ and H(z)…”

Section: 3mentioning

confidence: 99%