Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds

By Victor Beresnevich, V. Bernik, Dmitry Kleinbock, Gregory Margulis. 2002

101 citations recorded by the source corpus.

The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/77cadf0c240b2c6c6ec1b22a2556bf5e7bf2dbfb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Rational points near manifolds and metric Diophantine approximation (2009, background): “nes rather than rational points. 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, 11, 18] and very deep answers regarding the dual version of Problem 2.2 found [5, 6, 15, 21, 23]”
Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation (2005, methodology): “Combining the approach of the present paper with the methods of [BKM] and [BBKM] respectively, it seems plausible that both convergence and divergence cases can be proved for f ∗λ as in Theorem 1.”
Extremal subspaces and their submanifolds (2002, background): “(The case of convergence easily follows from the Borel-Cantelli Lemma.) It has been recently proved, see [BKM] for the convergence part and [ BBKM ] for the divergence part, that the same dichotomy takes place for any nondegenerate submanifold of Rn; in other words, for a smooth map f : U ! Rn which”
On fractal measures and diophantine approximation (2005, result): “Note that even for volume measures on proper nondegenerate submanifolds this problem is open, although there are definitive results in a dual (linear form) setting [BKM, Be, BBKM ].”
Rational points on the unit sphere (2008, background): “Problems of the latter type are addressed by the “Metric Diophantine Approximation on Manifolds” literature. (See, for example, [2], [ 1 ], [6].)”
An extension of quantitative nondivergence and applications to Diophantine exponents (2005, methodology): “The author expects this conjecture, as well as the multiplicative analogue of its convergence case, to be provable by a combination of the methods of [BKM, BBKM , G1,G3] and the present paper.”