关于Takagi函数水平集基数的分布
On the distribution of the cardinalities of level sets of the Takagi function
摘要：
设T是Takagi连续但无处可微函数。已知几乎所有水平集（关于T的值域上的勒贝格测度）都是有限的。我们表明，最常见的基数的水平集的T是两个，并详细调查的一组纵坐标y，使水平集在水平y正好有两个元素。作为一个副产品，我们得到了一个简单的迭代程序求解方程T（x）=y。我们进一步证明了任何正的偶数整数作为某个水平集的基数出现，并研究了当从T的范围内随机选取一个纵坐标y时，哪些基数以正概率出现。结果的关键是一个水平集的集合方程组，它是由T.这些集合方程产生了一个系统的水平集的基数之间的线性关系，在不同的水平，本文的所有结果流。
英文摘要：
Let T be Takagi's continuous but nowhere-differentiable function. It is known that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. We show that the most common cardinality of the level sets of T is two, and investigate in detail the set of ordinates y such that the level set at level y has precisely two elements. As a by-product, we obtain a simple iterative procedure for solving the equation T(x)=y. We show further that any positive even integer occurs as the cardinality of some level set, and investigate which cardinalities occur with positive probability if an ordinate y is chosen at random from the range of T. The key to the results is a system of set equations for the level sets, which are derived from the partial self-similarity of T. These set equations yield a system of linear relationships between the cardinalities of level sets at various levels, from which all the results of this paper flow.
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。
