下有理逼近与Farey阶梯
Lower rational approximations and Farey staircases
AI总结：
该研究定义了实数的下有理逼近及由其累积平均构成的Farey阶梯函数，证明该函数序列单调递增，明确了其极限行为并揭示了适当归一化下的分形结构。
摘要：
对于实数$x$，将$\frac1n \rfloor nx \backslash$称为$x$的第$n$个下有理逼近。我们研究了由$x$的前$n$个下有理逼近的累积平均定义的函数，将其命名为Farey阶梯函数。该函数序列是单调递增的。我们确定了这些函数的极限行为，并证明它们在适当归一化下呈现分形结构。
英文摘要：
For a real number $x$, call $\frac1n \lfloor nx \rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
