Level sets of the Takagi function: generic level sets Jeffrey LagariasZachary Maddock 26A4526A2726A3028A25Hausdorff dimensionLevel setSingular measureTakagi function 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4554 10.1512/iumj.2011.60.4554 en Indiana Univ. Math. J. 60 (2011) 1857 - 1884 state-of-the-art mathematics http://iumj.org/access/