论无理子空间的若干性质

On quadratic irrationals with bounded partial quotients

It is shown that for some explicit constants $c>0, A>0$, the asymptotic for the number of positive non-square discriminants $D<x$ with fundamental solution $\varepsilon_D< x^{\frac 12+α}, 0<α<c$, remains preserved if we require moreover $\mathbb Q(\sqrt D)$ to contain an irrational with partial quotients bounded by $A$.

On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers

In this paper we study the $b$-ary expansions of the square roots of the function defined by the recurrence $f_b(n)=b f_b(n-1)+n$ with initial value $f(0)=0$ taken at odd positive integers $n$, of which the special case $b=10$ is often referred to as the "schizophrenic" or "mock-rational" numbers. Defined by Darling in $2004$ and studied in more detail by Brown in $2009$, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases $…

Examples of badly approximable vectors over number fields

We consider approximation of vectors $\mathbf{z}\in F\otimes\mathbb{R}\cong\mathbb{R}^r\times\mathbb{C}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of $SL_2(\mathcal{O}_F)\backslash SL_2(F\otimes\mathbb{R})$ naturally associated to (totally indefinite, anisotropic) $F$-rational binary quadratic and Hermitian forms, a generalization of the well-known fact that quadratic irrationals are badly approximable over $\mathbb{Q}$.

A Multidimensional Gauss Map

The classical Gauss Map is a piecewise continuous map from the unit interval to itself. From this map we retrieve the continued fraction expansion of irrational numbers and its dynamical properties give information about some arithmetic and algebraic properties of irrational numbers. In this notes we will explore some generalizations of the Gauss Map to higher dimensions and pose some questions and conjectures about the arithmetic/algebraic information that these maps may carry.

$ L(2,(\frac d\cdot))$ 的快速收敛无理级数

Fast converging irrational series for $ L(2,(\frac d\cdot))$

本文基于 Guillera-Rogers 理论，利用 Kronecker 定理对格和线性组合进行研究，得到了多个 Dirichlet $L$-函数特殊值的几何收敛级数，其适用范围超越了 Guillera-Rogers 的可解情形。

关于Beatty序列的素数和

On sums of primes from Beatty sequences

设$k \ge 2$和$α_1, β_1, ..., α_k, β_k$是实数，使得$α_i$是无理数且大于1.进一步假设某个比率$α_i/α_j$是无理数。研究了整数$n$的$$ p_1 + p_2 + ... + p_k = n, $$形式的表示，其中$p_i$是Beatty序列$$ \mathcal B_i = \left\{n \in \mathbb N : n = [ α_i m + β_i ] \text{for some} m \in \mathbb Z \right\}. $$的素数

初等数论中的几个问题

Some conjectures in elementary number theory

我们宣布了一些与之相关的，并产生于研究素数和无理数在$\mathbb{R}$。所有这些都尽可能得到数字核查的支持。

某些显式的坏逼近对

Some explicit badly approximable pairs

考虑了一对无理数的具有公分母的超范数同时有理逼近的丢番图逼近问题，并对某些具有大逼近常数的无理数对进行了显式计算.其中一对是迄今为止计算出的最差的近似对。

Schmidt子空间定理的一些推广

Some generalizations of Schmidt's subspace theorem

本文给出了Schmidt子空间定理针对任意高次多项式族的定量版本，并针对射影代数簇中任意闭子概形族提出了该定理的推广。