Victor Beresnevich, Sanju Velani
In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a discussion on possible generalisations including a selection of various open problems.
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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.
This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed.
The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical properties of these exceptional sets are closely related to fundamental results in the metrical theory of Diophantine approximation. The counterpart of Diophantine approximation in hyperbolic space and a dynamical interpretation which led to the very general notion of `shrinking targets' are sketched and the recent use of flows in homogeneous spaces of lattice…
We give an elementary proof of a recent metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.
In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine-Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximating function.
In this paper we investigate the metrical theory of Diophantine approximation associated with linear forms that are simultaneously small for infinitely many integer vectors; i.e. forms which are close to the origin. A complete Khintchine--Groshev type theorem is established, as well as its Hausdorff measure generalization. The latter implies the complete Hausdorff dimension theory.
A uniform metrical theorem in multiplicative Diophantine approximation
本文研究了勒贝格通用二维实数对的乘性丢番图逼近问题，给出了满足特定增长条件下相关小值乘积数量的渐近公式。
Metric theory of inhomogeneous Diophantine approximations with a fixed matrix
本文针对固定矩阵情形，利用转移原理建立矩阵向量配对与转置矩阵丢番图性质的双向关联，证明齐次丢番图逼近经典结论的对应版本，解答相关公开问题，构建了对应的非齐次丢番图逼近度量理论。
The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.
Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.
We begin with a brief treatment of Hausdorff measure and Hausdorff dimension. We then explain some of the principal results in Diophantine approximation and the Hausdorff dimension of related sets, originating in the pioneering work of Vojtech Jarnik. We conclude with some applications of these results to the metrical structure of exceptional sets associated with some famous problems. It is not intended that all the recent developments be covered but they can be found in the references cited.
It is shown that for any translation invariant outer measure M, the M-measure of the intersection of any subset of R^n that is invariant under rational translations and which does not have full Lebesgue measure with an the closure of an open set of positive measure cannot be positive and finite. Analogues for $p$-adic fields and fields of formal power series over a finite field are established. The results are applied to some problems in metric Diophantine approximation.
The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main result is that, for almost every real number α, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|nα\|=0.
In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bmθ\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bmθ)= 1/…
Extreme value laws for intrinsic Diophantine approximation on spheres
本文研究球面上内在丢番图逼近的极值行为，证明在稀疏高度序列下逼近质量服从Weibull极限分布，计数函数渐近泊松。
We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois cohomology. The non-abelian reciprocity law then states that the global points are contained in the kernel of all the reciprocity maps.
Zero-one laws are a central topic in metric Diophantine approximation. A classical example of such laws is the Borel-Bernstein theorem. In this note, we prove a complex analogue of the Borel-Bernstein theorem for complex Hurwitz continued fractions. As a corollary, we obtain a complex version of Khinchin's theorem on Diophantine approximation.
Limit laws for cotangent and Diophantine sums
本文针对带幂奇点的1周期函数相关和式，证明其极限定律并给出收敛速度估计，结果适用于余切和与分数部分倒数和，主要工具包括Schmidt方法、Gauss-Kuzmin问题及$ψ$-混合随机变量理论。
Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of lattices under one-parameter unipotent subgroups of $\GL_2(\wh K)$ with approximation exponents and continued fraction expansions of elements of the fiel…
We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions.
Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
该研究推广了Kleinbock与Wadleigh提出的一致非齐次丢番图逼近零一律，引入任意权函数，基于转移原理给出简洁证明，刻画了带固定矩阵的$g$-Dirichlet对集合与坏逼近矩阵的性质，且相关测度结论适用于加权丢番图逼近。
Twisted Diophantine Approximation II: Uniform Theory
本文提出一种统一方法，利用对角格轨迹的逐次极小值精确计算扭曲一致丢番图逼近的Hausdorff维数，涵盖端点集、水平集及零满定律，并推广了多个已知结果。
Extreme values and logarithm laws for toral translations
该研究针对非齐次丢番图逼近中归一化第r近邻距离函数，基于齐次动力学，结合泊松近似策略与多重等分布结果，证明其极值极限定律及对应对数定律。
Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation
本文研究了加权和乘法Diophantine逼近中出现的limsup集的Lebesgue测度、Hausdorff测度和Fourier维度，提供了零-全律判定准则，并确定了这些集的精确Fourier维度。
Effective equidistribution for multiplicative Diophantine approximation on lines
本文通过证明 SL(3,R)/SL(3,Z) 中幂幺轨道的有效等分布，将 Littlewood 猜想在直线上几乎处处加强了两个对数因子，并利用对偶 Bohr 集理论改进 Kleinbock 的收敛结果及齐性空间对数律。
On a Kurzweil type theorem via ubiquity
本文针对非齐次丢番图逼近，在放宽不良逼近条件下，利用遍在系统构造证明了 Kurzweil 型定理的发散部分，并讨论其对应结果。
Zero-one laws for uniform approximation via Gaussian and Eisenstein integers
研究通过高斯整数商和艾森斯坦整数商对复数进行一致丢番图逼近的零一律，利用齐次动力学工具归结为收缩目标问题，通过不同论证得出临界轨迹邻域测度估计等新要素。
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