Dimension of Images of Large Level Sets

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 $0<α<1$ let $I_α(f)$ denote the set of values $y\in\mathbb{R}$ whose preimage $f^{-1}(y)$ has Hausdorff dimension $\dim f^{-1}(y) \ge α$. We consider how large can be the Hausdorff dimension of $I_α(f)$, 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 $\dim I_α(f)$ is $\displaystyle\frac{1-α}k$.

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