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Anish Ghosh - One of the best experts on this subject based on the ideXlab platform.
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A Khintchine–GROSHEV THEOREM FOR AFFINE HYPERPLANES
International Journal of Number Theory, 2011Co-Authors: Anish GhoshAbstract:We prove the divergence case of the Khintchine–Groshev theorem for a large class of affine hyperplanes, completing the convergence case proved in [11] and answering in part a question of Beresnevich et al. ([4]). We use the mechanism of regular systems developed in [4] and estimates from [11].
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a Khintchine groshev theorem for affine hyperplanes
International Journal of Number Theory, 2011Co-Authors: Anish GhoshAbstract:We prove the divergence case of the Khintchine–Groshev theorem for a large class of affine hyperplanes, completing the convergence case proved in [11] and answering in part a question of Beresnevich et al. ([4]). We use the mechanism of regular systems developed in [4] and estimates from [11].
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Diophantine exponents and the Khintchine Groshev theorem
Monatshefte für Mathematik, 2010Co-Authors: Anish GhoshAbstract:We prove the convergence case of the Khintchine–Groshev theorem for affine subspaces and their nondegenerate submanifolds, answering a conjecture of D. Kleinbock.
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A Khintchine type theorem for hyperplanes
arXiv: Number Theory, 2005Co-Authors: Anish GhoshAbstract:We prove that the convergence Khintchine theorem holds for affine hyperplanes whose parametrizing matrices satisfy a mild Diophantine condition. We use the dynamical method of Kleinbock-Margulis.
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A Khintchine type theorem for hyperplanes
Journal of the London Mathematical Society, 2005Co-Authors: Anish GhoshAbstract:The convergence case of a Khintchine-type theorem for a large class of hyperplanes is obtained. The approach to the problem is from a dynamical viewpoint, and a method due to Kleinbock and Margulis is modified to prove the result.
Victor Beresnevich - One of the best experts on this subject based on the ideXlab platform.
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diophantine approximation on manifolds and the distribution of rational points contributions to the convergence theory
International Mathematics Research Notices, 2016Co-Authors: Victor Beresnevich, R C Vaughan, Sanju Velani, Evgeniy ZorinAbstract:In this article, we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold M ⊂ R^n is of dimension strictly greater than (n + 1)/2 and satisfies a natural non-degeneracy condition, then M is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.
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diophantine approximation on manifolds and the distribution of rational points contributions to the convergence theory
arXiv: Number Theory, 2015Co-Authors: Victor Beresnevich, R C Vaughan, Sanju Velani, Evgeniy ZorinAbstract:In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.
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classical metric diophantine approximation revisited the Khintchine groshev theorem
International Mathematics Research Notices, 2009Co-Authors: Victor Beresnevich, Sanju VelaniAbstract:Let denote the set of ψ-approximable points in . Under the assumption that the approximating function ψ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of . The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on ψ 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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Classical Metric Diophantine Approximation Revisited: The Khintchine–Groshev Theorem
International Mathematics Research Notices, 2009Co-Authors: Victor Beresnevich, Sanju VelaniAbstract:Let denote the set of ψ-approximable points in . Under the assumption that the approximating function ψ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of . The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on ψ 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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classical metric diophantine approximation revisited the Khintchine groshev theorem
arXiv: Number Theory, 2008Co-Authors: Victor Beresnevich, Sanju VelaniAbstract: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 the set of $\psi$-approximable matrices in $\R^{mn}$. 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 $n > 2$ (Sprindzuk) or when $n=1$ and $m>1$ (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 multi-dimensional analogue of Catlin's Conjecture.
Wei Sun - One of the best experts on this subject based on the ideXlab platform.
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extensions of levy Khintchine formula and beurling deny formula in semi dirichlet forms setting
Journal of Functional Analysis, 2006Co-Authors: Wei SunAbstract:Abstract The Levy–Khintchine formula or, more generally, Courrege's theorem characterizes the infinitesimal generator of a Levy process or a Feller process on R d . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Levy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.
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Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting
Journal of Functional Analysis, 2006Co-Authors: Wei SunAbstract:Abstract The Levy–Khintchine formula or, more generally, Courrege's theorem characterizes the infinitesimal generator of a Levy process or a Feller process on R d . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Levy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.
Daniel C. Alvey - One of the best experts on this subject based on the ideXlab platform.
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A Khintchine type theorem for affine subspaces
International Journal of Number Theory, 2020Co-Authors: Daniel C. AlveyAbstract:We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of [Formula: see text] intersected with an appropriate subspace is also obtained.
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A Khintchine type theorem for affine subspaces
arXiv: Number Theory, 2020Co-Authors: Daniel C. AlveyAbstract:We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parametrizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine's theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of W(\tau) intersected with an appropriate subspace is also obtained.
M. M. Dodson - One of the best experts on this subject based on the ideXlab platform.
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Metrical Diophantine approximation for quaternions
arXiv: Number Theory, 2014Co-Authors: M. M. Dodson, Brent EverittAbstract:Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.
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The Khintchine–Groshev theorem for planar curves
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 1999Co-Authors: Victor Beresnevich, V. I. Bernik, H. Dickinson, M. M. DodsonAbstract:The analogue of the classical Khintchine–Groshev theorem, a fundamental result in metric Diophantine approximation, is established for smooth planar curves with non–vanishing curvature almost everywhere.
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the Khintchine groshev theorem for planar curves
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 1999Co-Authors: Victor Beresnevich, V. I. Bernik, H. Dickinson, M. M. DodsonAbstract:The analogue of the classical Khintchine–Groshev theorem, a fundamental result in metric Diophantine approximation, is established for smooth planar curves with non–vanishing curvature almost everywhere.
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A Khintchine-type version of Schmidt's theorem for planar curves
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 1998Co-Authors: V. I. Bernik, H. Dickinson, M. M. DodsonAbstract:An analogue of the convergence part of the Khintchine–Groshev theorem is proved for planar curves obeying certain curvature conditions.
