实部为二分之一的复数丢番图逼近的拉格朗日谱
Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half
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中文总结 AI 辅助
该研究在A. Schmidt相关定理启发下，刻画实部为1/2的复数的受限拉格朗日谱，推导其分形性质并证明满足特定逼近下界的复数集不可数。
中文摘要 AI 辅助
受A. Schmidt关于复拉格朗日谱中小于2部分的定理启发，我们研究受限拉格朗日谱$L_{\frac{1}{2}+i\mathbb{R}}$，该谱源于对形如$\frac{1}{2}+i\alpha$（$\alpha\in\mathbb{R}\setminus\mathbb{Q}$）的复数用高斯有理数$p/q$（$p,q\in\mathbb{Z}[i]$，$q\neq0$）进行逼近。我们证明该谱可通过与实马蹄映射相关的动力学谱描述，进而得到$L_{\frac{1}{2}+i\mathbb{R}}$的若干分形性质，如维数函数$t\mapsto\dim_H(L_{\frac{1}{2}+i\mathbb{R}}\cap(-\infty,t))$的连续性。我们还证明，对所有$p,q\in\mathbb{Z}[i]$、$q\neq0$，满足$\left|z-\frac{p}{q}\right|\geq\frac{1}{2|q|^2}$的复数集是不可数的，且该不等式对所有形如$z=\frac{1}{2}(1+i\theta)$（$\theta\in\mathbb{R}\setminus\mathbb{Q}$，为Schmidt的C-极小型之一的根）的复数成立。
英文摘要
Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\frac{1}{2}+i\mathbb{R}}$ arising from the approximation of complex numbers of the form $\frac{1}{2}+iα$, $α\in\mathbb{R}\setminus\mathbb{Q}$ by Gaussian rationals $p/q$ with $p,q\in\mathbb{Z}[i]$, $q\neq 0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\frac{1}{2}+i\mathbb{R}}$, such as continuity of the dimension function $t\mapsto\dim_H(L_{\frac{1}{2}+i\mathbb{R}}\cap(-\infty,t))$. We also prove that the set of complex numbers $z$ satisfying \begin{equation*} \left\lvert z-\frac{p}{q}\right\rvert\geq\frac{1}{2|q|^2}, \quad\text{for all } p,q\in\mathbb{Z}[i], q\neq 0, \end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\frac{1}{2}(1+iθ)$ where $θ\in\mathbb{R}\setminus\mathbb{Q}$ is a root of one of Schmidt's $C$-minimal forms.
发表机构
- IMPA（巴西国家纯数学与应用数学研究所）
- SUSTech International Center for Mathematics（南方科技大学国际数学中心）
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