Weak tangents and level sets of Takagi functions

TL;DR

This paper investigates the properties of Takagi functions and their level sets, revealing conditions under which large level sets exist and demonstrating that their graphs can have complex fractal dimensions.

Contribution

It establishes a link between parameters of Takagi functions and the existence of large level sets, also analyzing their fractal dimensions and weak tangents.

Findings

Existence of large level sets when parameters satisfy specific polynomial roots.

Graphs of certain Takagi functions have Assouad dimension exceeding upper box dimension.

Construction of weak tangents with Hausdorff dimension larger than the box dimension.

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.