Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

Victor Beresnevich; Sanju Velani

doi:10.1093/imrn/rnp119

Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

Victor Beresnevich, Sanju Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let A(n,m)(psi) denote the set of psi-approximable points in R-mn. Under the assumption that the approximating function psi is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

Original language	English
Pages (from-to)	69-86
Number of pages	18
Journal	International Mathematics Research Notices
Volume	2010
Issue number	1
DOIs	https://doi.org/10.1093/imrn/rnp119
Publication status	Published - 2010

Keywords

SCHAEFFER CONJECTURE
DUFFIN
LAWS

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10.1093/imrn/rnp119

http://arxiv.org/abs/0811.0809

Cite this

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title = "Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem",

abstract = "Let A(n,m)(psi) denote the set of psi-approximable points in R-mn. Under the assumption that the approximating function psi is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.",

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T1 - Classical Metric Diophantine Approximation Revisited

T2 - The Khintchine-Groshev Theorem

AU - Beresnevich, Victor

AU - Velani, Sanju

PY - 2010

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N2 - Let A(n,m)(psi) denote the set of psi-approximable points in R-mn. Under the assumption that the approximating function psi is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

AB - Let A(n,m)(psi) denote the set of psi-approximable points in R-mn. Under the assumption that the approximating function psi is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

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KW - DUFFIN

KW - LAWS

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U2 - 10.1093/imrn/rnp119

DO - 10.1093/imrn/rnp119

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SP - 69

EP - 86

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 1

ER -

Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

Abstract

Keywords

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