论文标题
部分可观测时空混沌系统的无模型预测
A uniform metrical theorem in multiplicative Diophantine approximation
论文作者
论文摘要
对于lebesgue通用$（x_1，x_2）\ in \ mathbb {r}^2 $，我们研究了产品的小值$ q \ cdot \ | qx_1 \ | \ cdot \ | qx_2 \ | $ with $ q \ in \ mathbb {n} $，其中$ \ | \ cdot \ | $表示与最接近整数的距离。主要结果给出了$ 1 \ le q \ le t $的渐近公式，以便$$ a_t a_t <q \ cdot \ | qx_1 \ | \ cdot \ | qx_2 \ | \ leq b_t \ quad \ textrm {and} \ quad \ | qx_1 \ |，\ | qx_2 \ | | \ | \ | \ | \ leq c_t $$用于给定序列$ a_t，b_t，b_t，b_t，c_t $满足某些增长条件。
For Lebesgue generic $(x_1,x_2)\in \mathbb{R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q\le T$ such that $$ a_T <q\cdot \|qx_1\| \cdot \|qx_2\|\leq b_T \quad \textrm{and} \quad \|qx_1\|, \|qx_2\|\leq c_T $$ for given sequences $a_T,b_T, c_T$ satisfying certain growth conditions.
