A pathological construction for real functions with large collections of level sets

TL;DR

This paper constructs real functions with level sets whose Hausdorff dimensions can be arbitrarily close to 1, showing the maximal possible size of such collections even for differentiable functions.

Contribution

It demonstrates that the Hausdorff dimension of the collection of level sets with maximal dimension can be as large as 1, even for functions with finite differentiability, establishing a sharp bound.

Findings

Hausdorff dimension of level sets can approach 1

Maximal collection size is achievable for differentiable functions

Sharp bound for Hausdorff dimension of level set collections

Abstract

Consider all the level sets of a real function. We can group these level sets according to their Hausdorff dimensions. We show that the Hausdorff dimension of the collection of all level sets of a given Hausdorff dimension can be arbitrarily close to 1, even if the function is differentiable to some level. By definition of Hausdorff dimension it is clear, for any real function $f(x)$ and any $\alpha \in [0,1]$, that $\dim_{H} \left\{ {0.03in} y \ : \ \dim_{H} (f^{-1}(y)) \geq \alpha {0.03in} \right\} \leq 1$. What is surprising, and what we show, is that this is actually a sharp bound. That is, $$\sup \left\{ {0.03in} \dim_{H} \left\{ {0.03in} y \ : \ \dim_{H} (f^{-1}(y)) = 1 {0.03in} \right\} \ : \ f \in C^{k} {0.03in} \right\} = 1,$$ for any $k \in \mathbb{Z}_{\geq 0}$.