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Bernik, V. I.; Kleinbock, Dmitry; Margulis, G. A.

doi:10.1155/s1073792801000241

Cited by 87 publications

(3 citation statements)

References 12 publications

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“…The problems in the dual case are much more tractable and progress has been significantly better. In particular, the dual version of Problem 2.1 has been fully settled [3], [6], [19] and very deep answers regarding the dual version of Problem 2.2 found [5], [7], [16], [22], [24]. However, as we shall see, Problems 2.1 and 2.2 (nondual) have more or less been understood only in R 2 .…”

Section: Diophantine Approximation On Manifoldsmentioning

confidence: 96%

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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Section: Diophantine Approximation On Manifoldsmentioning

confidence: 96%

Rational points near manifolds and metric Diophantine approximation

Beresnevich

2012

Ann. Math.

145

View full text Add to dashboard Cite

show abstract

“…В работе Бересневича [6] было показано, что в аналоге неравенства (3) для действительных чисел условие монотонности функции также можно опустить. При этом на последнем этапе существенно использовалась теорема Берника-Клейнбока-Маргулиса [7]. В данной работе доказывается аналог теоремы Бересневича для комплексных чисел.…”

Section: Doi: 104213/mzm4612unclassified

“…Зафиксируем H такое, что 2 t H < 2 t+1 . Число значений H, удовлетворяющих (7), не превосходит 2 tθ1 . Если квадрат со стороной 2 −tσ3 , σ 3 > 0, содержит не более 3 · 2 tθ1 многочленов P ∈ M t (l), то для меры множества B t = P ∈Mt(l) B(P ) получим следующую оценку:…”

Section: доказательство теоремыunclassified