高木函数的水平集有多大？
How large are the level sets of the Takagi function?
摘要：
设T是Takagi连续但无处可微函数。本文从概率的角度和Baire范畴的角度研究了T的水平集的大小。我们首先给出三个最近发表的结果更基本的证明。第一种是由于Z. Buczolich指出，几乎所有的水平集（关于T范围上的勒贝格测度）都是有限的。第二种是J. Lagarias和Z. Maddock指出，水平集中的平均点数是无限的。第三个结果，也是由Lagarias和Maddock得出的，指出水平集中包含的局部水平集的平均数量是3/2。在第二部分的文件中，它表明，在上述结果相比，一组纵坐标y与不可数无限水平集是剩余的，并给出了一个相当明确的描述这一集。本文还对Lagarias和Maddock的一个问题给出了否定的回答，证明了大多数水平集（在Baire范畴的意义下）包含无穷多个局部水平集，并且水平集的连续统甚至包含不可数个局部水平集.最后，几个主要的结果被扩展到一个版本的T与任意符号的被加数。
英文摘要：
Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.
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