Dimension of Images of Large Level Sets

TL;DR

This paper investigates the maximum Hausdorff dimension of the set of values for which the preimage under a smooth function has a Hausdorff dimension at least alpha, establishing a sharp upper bound of (1-alpha)/k.

Contribution

It provides a precise upper bound on the Hausdorff dimension of the set of such values for k-times differentiable functions, extending understanding of level set structures.

Findings

Maximum Hausdorff dimension of I_alpha(f) is (1-alpha)/k.

The bound is sharp and attained by certain functions.

Results apply to functions with positive-length domains.

Abstract

Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<\alpha<1$ let $I_\alpha(f)$ denote the set of values $y\in\mathbb{R}$ whose preimage $f^{-1}(y)$ has Hausdorff dimension $\dim f^{-1}(y) \ge \alpha$. We consider how large can be the Hausdorff dimension of $I_\alpha(f)$, as $f$ ranges over the set $C^k(E,\mathbb{R})$ of all $k$-times continuously-differentiable functions from $E$ into $\mathbb{R}$. We show that the sharp upper bound on $\dim I_\alpha(f)$ is $\displaystyle\frac{1-\alpha}k$.