Multiplicative Diophantine exponents of hyperplanes and their nondegenerate submanifolds

Extremal subspaces and their submanifolds

It is known that the properties of almost all points of R^n being not very well (multiplicatively) approximable are inherited by nondegenerate in R^n (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper affine subspaces, and prove that the aforementioned diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional diophantine approximation and dynamics of lattices in Euclidean spaces.

Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation

The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.

欣钦定理与流形上的丢番图逼近

Khintchine's theorem and Diophantine approximation on manifolds

本文提出结合几何、动力学思想与“通有部分和特殊部分”新技术的新方法，证明了$\mathbb{R}^n$任意非退化子流形的收敛型欣钦型定理，细化得到相关豪斯多夫维数精确值，解决了丢番图逼近领域的长期问题。

Metric Diophantine approximation for systems of linear forms via dynamics

The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong extremality' of arbitrary finite collection of smooth nondegenerate submanifolds of ${\bold R}^n$. The proofs are based on generalized quantitative nondivergence estimates for translates of measures on the space of lattices.

Diophantine approximations on random fractals

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable …

双p-进迪奥菲安逼近在流形上

Dual $p$-adic Diophantine approximation on manifolds

本文研究了双p-进逼近在流形上的收敛性问题，针对维度至少为三的超曲面及部分满足条件的流形，证明了同质p-进收敛结果，并在非同质情况下提供了较弱的结论。

Quantitative Khintchine in Simultaneous Approximation

In a ground-breaking work \cite{BY}, Beresnevich and Yang recently proved Khintchine's theorem in simultaneous Diophantine approximation for nondegenerate manifolds resolving a long-standing problem in the theory of Diophantine approximation. In this paper, we prove an effective version of their result.

非齐次坏逼近向量的胜出性与零测性

Winning and nullity of inhomogeneous bad

本文证明了加权非齐次坏可逼近向量的超平面绝对胜出性，并利用对偶性与齐次动力学定量非发散估计证明非退化曲线和解析流形上几乎处处不具有该性质。

逼近向量的几何与算术性质

Geometric and arithmetic aspects of approximation vectors

该研究探讨丢番图逼近中与逼近向量相关的三类几何、算术量的渐近分布，通过阿代尔空间单参数流截面等分布方法，得到几乎所有实数向量及全实数域情形下的极限测度，推广了前人结果至高维与联合等分布场景。