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arXiv · 2010.11556

Dimension of Images of Large Level Sets

Anthony G. O'Farrell·Gavin Armstrong
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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 $\displaystyle\frac{1-\alpha}k$.

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Anthony G. O'Farrell, Gavin Armstrong. 2020-10-22. Dimension of Images of Large Level Sets. https://doi.org/10.7146/math.scand.a-129246

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