关于同时丢番图逼近中某些极限集的豪斯多夫维数的注记

迭代系统和一般系统的丢番图性质：数量近似性和连通性

Diophantine properties of IETs and general systems: Quantitative proximality and connectivity

我们提出了一般系统的收缩目标的结果，重点是应用程序的间隔交换变换（$(J,T)$），$J=[0,1)$。特别地，我们证明了，如果IET $(J,T)$是遍历（相对于勒贝格测度$\lam$），那么等式\[ \liminf_{n\to\infty}\limits n |T^n(x)-y|=0 \tag{A1} \]对$\lam\ttimes\lam$ -a成立。a. $(x,y)\in J^2$。遍历性假设是必要的：结果并不延伸到所有的最小迭代。（A1）中的因子$n$是最佳的（e.例如，在一个实施例中，它不能被$n \ln(\ln(\ln n))$取代。另一方面，对于勒贝格几乎所有的3-INUS $(J,T)$，我们证明，对于所有的$\eps>0$ \[ \liminf_{n\to\infty}\limits n^\eps |T^n(x)-T^n(y)|= \infty,\quad \text{for Lebesgue a. a.} (x,y)\in J^2. \tag{A2} \]这应该是对比的平等$ \liminf_{n\to\infty}\limits |T^n(x)-T^n(y)|=0, $的a。a. $(x,y)\in J^2$，它成立，因为$(J^2, T\times T)$是遍历的（因为一般的3-整数$(J,T)$是弱混合的）。我们还证明了没有3-IET是强拓扑混合的。

关于混合Littlewood猜想的度量考虑

Metric considerations concerning the mixed Littlewood Conjecture

本文的主要目的是在de Mathan-Teulie猜想（也称为“混合Littlewood猜想”）的框架内建立一个丢番图逼近的度量理论。设p为素数。我们的主要结果的一个推论是，对几乎所有实数α，\liminf_{n\rar\infty} n（\log n）^2 |n| _p n α =0。

非齐次丢番图逼近中的豪斯多夫维数

Hausdorff dimension in inhomogeneous Diophantine approximation

设$α$为无理数实数。我们证明了$ε$ -差逼近数\[ \mathrm{Bad}^\varepsilon (α) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| qα-x \| \geq \varepsilon \} \]的集合对于某个正的$ε$具有满豪斯多夫维数当且仅当$α$平均奇异.该条件等价于$α$的偏导数$a_i$的平均值$\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$随$k$趋于无穷大。我们还考虑单侧逼近，当$a_i$趋于无穷大时得到了更强的结果，并在高维中建立了部分结果。

On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_q ψ(q)=\infty$ and any number $γ,$ the following set \[ W(ψ,γ)=\{x\in [0,1]: |qx-p-γ|< ψ(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if $γ$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced …

Levy-Khintchin Theorem for best simultaneous Diophantine approximations

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_nθ, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real number $θ$ by the ordinary continued fraction algorithm. Beside these two main results, we show that when $d\ge2$, for almost…

On the Diophantine equation $σ_{2}(\overline{X}_{n})=σ_{n}(\overline{X}_{n})$

In this note we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of $σ_{2}(\overline{X}_{n})$ and $σ_{n}(\overline{X}_{n})$ together with the biggest value of the variable $x_{n}$ appearing in the solution. Moreover, we enumerate all solutions for $n\leq 16$ and discuss the set of values of $x_{n}/x_{n-1}$ over elements of $S(n)$.

欧几里得范数下同时最佳丢番图逼近分母的增长性

On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product

该研究针对n≥2维欧几里得范数下的同时最佳丢番图逼近，证明了其分母序列的增长下界，推导得到gₙ(α)、G(n)的下界及underline{D}ₙ(α)的上界，给出了n=2、3时的具体数值结果。

关于Kronecker丢番图定理的局部化

On localization in Kronecker's diophantine theorem

研究了克罗内克丢番图定理中的局部化问题。我们引入了一个概率的方法，允许扩展一般$\Q$线性无关序列的结果图兰有关的序列$ (\log p_\ell)$，$p_\ell$是$\ell$第素数。

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。

Dynamical Diophantine Approximation Exponents in Characteristic $p$

Let $ϕ(z)$ be a non-isotrivial rational function in one-variable with coefficients in $\overline{\mathbb{F}}_p(t)$ and assume that $γ\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ is not a post-critical point for $ϕ$. Then we prove that the diophantine approximation exponent of elements of $ϕ^{-m}(γ)$ are eventually bounded above by $\lceil d^m/2\rceil+1$. To do this, we mix diophantine techniques in characteristic $p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we tak…

全局函数域上非齐次丢番图逼近中的 Hausdorff 维数

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

本文研究全局函数域完备化上的非齐次坏逼近集合，利用齐次动力学的有效熵刚性给出其 Hausdorff 维数上界，并以平均奇异性条件刻画满维数情形。

分形测度的分解结果及其在迪奥菲内逼近中的应用

Disintegration results for fractal measures and applications to Diophantine approximation

本文研究了分形测度的分解结果，并应用于迪奥菲内逼近，证明了自相似测度和仿射不可约自相似测度在特定条件下的性质。

Dimension estimates for badly approximable affine forms

For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approximable matrices for fixed target $b$ and the set of $ε$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $ε$-badly appr…

关于方程 $a!b!=c!$ 中整数 $a$ 的稀疏性

On the sparsity of integers $a$ in solutions to $a!b!=c!$

研究方程 $a!b!=c!$ 中整数 $a$ 的稀疏性，证明其在渐进行密度上为零。

不等式逼近的维度理论与矩阵序列

Dimension theory of inhomogeneous Diophantine approximation with matrix sequences

本文研究了由矩阵序列生成的非齐次良好逼近点集的Hausdorff维度，统一了收缩目标集和回归集问题，并给出了显式的维度公式。