Weak tangents and level sets of Takagi functions

Abstract: In this paper we study some properties of Takagi functions and their level sets. We show that for Takagi functions $T_{a,b}$ with parameters $a,b$ such that $ab$ is a root of a Littlewood polynomial, there exist large level sets. As a consequence we show that for some parameters $a,b$, the Assouad dimension of graphs of $T_{a,b}$ is strictly larger than their upper box dimension. In particular we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.