article

Title: Level sets of the Takagi function: generic level sets

Authors: Zachary Maddock and Jeffrey C Lagarias

Issue: Volume 60 (2011), Issue 6, 1857-1884

Abstract:

$The Takagi function $\tau:[0,1]\to[0,1]$ is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets $L(y) = \{x:\tau(x) = y\}$ of the Takagi function $\tau(x)$. It shows that for a full Lebesgue measure set of ordinates $y$, these level sets are finite sets, but whose expected number of points is infinite. Complementing this, it shows that the set of ordinates $y$ whose level set has positive Hausdorff dimension is itself a set of full Hausdorff dimension $1$ (but Lebesgue measure zero). Finally, it shows that the level sets have a nontrivial Hausdorff dimension spectrum. The results are obtained using a notion of "local level set" introduced in a previous paper, along with a singular measure parameterizing such sets.$