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How large are the level sets of the Takagi function?
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.Comment: Added a new Section 5 with generalization of the main results; some
new and corrected proofs of the old material; 29 pages, 3 figure