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

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 $k \in \mathbb{Z}_{\geq 0}$.