Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions

连分数理论中 Dirichlet 不可改进数的收敛指数

Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions

本文研究连分数框架下 Dirichlet 不可改进数与收敛指数水平集的交集，对任意参数确定了集合 $G(\alpha,\beta)$ 与 $E(\alpha,\beta)$ 的 Hausdorff 维数公式。

Hausdorff dimension of a set in the theory of continued fractions

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.

连分数中部分商乘积的度量性质与几何平均

Metrical properties of the product of partial quotients with geometric mean in continued fractions

本文研究连分数中部分商乘积的几何平均的度量性质，分析了当指数呈线性间隔且乘积长度增长时，豪斯多夫维数的变化，发现维数随 $d$ 增大而增大，$t$ 不影响维数。

在 $L_p$ 范数下的 Dirichlet 可改进性

Dirichlet improvability in $L_p$-norms

通过分类 $L_p$ 单位球的临界格，用正则连分数展开中的模式完整刻画了 $L_p$ 范数下的 Dirichlet 可改进数集，解决了 Kleinbock 和 Rao 的两个公开问题。

Simultaneous Diophantine approximation on affine subspaces and Dirichlet improvability

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the base point of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence. We also prove a partial analogue regarding the Hausdorff measure theory. Furthermore, we obtain various results relating weighted Diophantine approximation and Dirichlet improvability. In particu…

A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions

We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions…

Weighted uniform Diophantine approximation of systems of linear forms

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an $m \times n$ matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb R})/\mathrm{SL}_{m+n}({\mathbb Z})$ avoiding a compact subset of the space of lattices called the critical locus defined with respect to …

Bad 减去范数 Dirichlet 的二分类现象

A dichotomy phenomenon for Bad minus normed Dirichlet

本文证明关于范数 Dirichlet 可改进数与坏逼近数交集的一个二分类现象：要么坏逼近数集合包含于该交集，要么其差集具有完全 Hausdorff 维数，并给出动力学解释。

Diophantine approximation on curves

Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $Ψ$-approximable points on nondegenerate manifolds. The problem relates the `size' of the set of $Ψ$-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any nondegenerate manifold. The convergence case, however, represents a major challenging open problem and progress thus far has been effectuated i…

抽象有理数的下极限逼近集

Liminf approximation sets for abstract rationals

本文研究抽象有理数框架下指数增长序列的下极限逼近集，对形如 $q^{-τ_i}$ 的加权逼近函数计算其 Hausdorff 维数，推广了 Jarník-Besicovitch 定理。

The Baker-Schmidt problem for dual approximation and some classes of functions

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff $f$-measure of the set of $Ψ$-approximable points on a nondegenerate manifold. We refine and extend our previous work [Int. Math. Res. Not. IMRN 2021, no. 12, 8845--8867] in which we settled the problem (for dual approximation) for hypersurfaces. We verify the GBSP for certain classes of nondegenerate submanifolds of codimension greater than $1$. Concretely, for codimension two or three, we provide examples of manifolds where the dependent variables can be chosen as quadratic forms. Our method requires the manifold to have eve…

$β$ -动力系统丢番图性质的二分法

A dichotomy law for the Diophantine properties in $β$-dynamical systems

设$β>1$是一个实数，并通过$T_β:x\mapsto βx\bmod 1$定义$β>1$上的$β$ -变换.此外，定义$$W_y(T_β,Ψ):=\{x\in [0, 1]:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\}$$和$$W(T_β,Ψ):=\{(x, y)\in [0, 1]^2:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\},$$，其中$Ψ:\mathbb{N}\to\mathbb{R}_{>0}$是正函数，使得$Ψ(n)\to 0$为$n\to \infty$。在本文中，我们证明了上述集合中的每一个都服从Jarnik型二分法，即广义Hausdorff测度要么为零，要么为满，这取决于某个级数的收敛或发散。这项工作完成了这些集的计量理论。

关于矩阵的Folklore集与Dirichlet谱

On the Folklore set and Dirichlet spectrum for matrices

本文研究$\mathbb R^{m\times n}$中既非奇异也非坏逼近的Dirichlet可改进矩阵的Folklore集，证明除特定整数对外其非空，构造具预设Dirichlet常数的子集以提供Dirichlet谱信息，并得豪斯多夫维数下界，补充了先前研究。

分形地毯的测度刚性与等分布

Measure rigidity and equidistribution for fractal carpets

研究由有限个收缩有理仿射膨胀生成的随机游走的平稳伯努利测度，证明在无质量逃逸条件下，沿发散对角矩阵序列的推进测度等分布，并推广到加权坏逼近向量和Dirichlet可改进向量的零测性质。