Step roots of Littlewood polynomials and the extrema of functions in the Takagi class

Abstract

We give a new approach to characterizing and computing the set of global maximizers and minimizers of the functions in the Takagi class and, in particular, of the Takagi--Landsberg functions. The latter form a family of fractal functions $f_{\alpha }:[0,1]\to \mathbb{R}$ parameterized by $\alpha \in (-2,2)$. We show that $f_{\alpha }$ has a unique maximizer in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi--Landsberg functions with $\alpha \in (-2,1/2]\cup (1,2)$. For $(1/2,1]$, we show that the step roots are dense in that interval. If $\alpha \in (1/2,1]$ is a step root, then the set of maximizers of $f_{\alpha }$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$, where $n$ is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi--Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup [1/2,2]$.