Simultaneous inhomogeneous diophantine approximation on manifolds
Simultaneous inhomogeneous diophantine approximation on manifolds
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流形上的联立非齐次丢番图近似
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
S. Velani
中科院分区：
文献类型：
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作者：
V. Beresnevich;S. Velani
在1998年，Kleinbock和Margulis证明了Sprindzuk猜想与度量丢番图近似有关（实际上是更强的Baker-Sprindzuk猜想）。本质上，该猜想指出，对于$$ {\mathbb{R}^n} $$的非退化子流形$$ \mathcal{M} $$上的几乎每个点x，同时齐次丢番图指数w 0（x）= 1/n。本文建立了Sprindzuk猜想的同时非齐次模拟。更精确地说，对于任何“非齐次”向量θ ∈ $ {\mathbb{R}^n} $$，我们证明了同时非齐次丢番图指数w 0（x，θ）对于几乎$ \mathcal{M} $$上的每个点x都是1/n。关键结果是一个非齐次转移原理，它使我们能够推导出齐次指数w 0（x）对于几乎所有x ∈ $$ \mathcal{M} $$为1/n当且仅当，对于任何θ ∈ $$ {\mathbb{R}^n} $$，非齐次指数w 0（x，θ）对于几乎所有x ∈ $$ \mathcal{M} $$= 1/n。本文介绍的非齐次迁移原理是我们最近发现的非齐次迁移原理的一个极其简化的版本。然而，应该强调的是，简化版本具有很大的优势，它将主要思想带到了最前沿，同时省略了抽象和技术概念，这些概念是在描述非均匀移情原理的所有荣耀时出现的。
In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine exponent w0(x) = 1/n for almost every point x on a nondegenerate submanifold $$ \mathcal{M} $$ of $$ {\mathbb{R}^n} $$. In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any “inhomogeneous” vector θ ∈ $$ {\mathbb{R}^n} $$ we prove that the simultaneous inhomogeneous Diophantine exponent w0(x,θ) is 1/n for almost every point x on $$ \mathcal{M} $$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w0(x) is 1/n for almost all x ∈ $$ \mathcal{M} $$ if and only if, for any θ ∈ $$ {\mathbb{R}^n} $$, the inhomogeneous exponent w0(x,θ) = 1/n for almost all x ∈ $$ \mathcal{M} $$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.