Pierce展开中Shallit闰年律的上下波动
Upper and lower fluctuations in Shallit's law of leap years in Pierce expansions
- Dankook University（檀国大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明Pierce展开中Shallit闰年律的上下极限由前n位数字乘积的对数下极限决定，并刻画了相应点集的稠密性与Hausdorff维数。
AI中文摘要：
Pierce展开中的Shallit闰年律给出了，对于Lebesgue几乎所有的$x\in[0,1]$，归一化差值$Nx$与由$x$的Pierce展开数字确定的截至第$N$年的闰年数之间的上极限和下极限。本文证明，对于每个$x\in[0,1]$，这两个极限均由$x$的前$n$个数字乘积的归一化对数的下极限决定。特别地，上极限和下极限总是具有相同的绝对值。作为应用，我们刻画了具有给定上极限和下极限的点集，并证明这些集合中的每一个，若非空，则在$[0,1]$中稠密，并且其与$[0,1]$的任何非空开子集的交集具有完整的Hausdorff维数。我们还将这些集合与由数字增长率定义的集合进行比较，并证明对于每个有限参数，它们的差集具有完整的Hausdorff维数。
英文摘要：
Shallit's law of leap years in Pierce expansions gives, for Lebesgue-almost every $x\in[0,1]$, the upper and lower limits of a normalized difference between $Nx$ and the number of leap years up to the year $N$ determined by the Pierce expansion digits of $x$. In this paper, we show that, for every $x\in[0,1]$, both limits are determined by the lower limit of a normalized logarithm of the product of the first $n$ digits of $x$. In particular, the upper and lower limits always have the same absolute value. As applications, we characterize the sets of points with prescribed upper and lower limits, and show that each of these sets, if non-empty, is dense in $[0,1]$ and its intersection with any non-empty open subset of $[0,1]$ has full Hausdorff dimension. We also compare these sets with the sets defined by the growth rate of the digits, and show that, for each finite parameter, their difference has full Hausdorff dimension.
