Dimension of Images of Large Level Sets

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 $\frac{1-\alpha }{k}$.

11 pages, 3 figures. Inaccuracies in the historical survey section 1.5 (second paragraph) have been corrected, and an acknowledgement added to M.Elekes, who pointed them out to me

References in corpus (2)

• Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$
• On the packing measure of slices of self-similar sets