Dimension of images of large level sets - Armstrong, Gavin; O'Farrell, Anthony G. (MATHEMATICA SCANDINAVICA, 2022)
Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\colon E→ \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 at least $α$. We consider how large can be the Hausdorff dimension of $I_α (f)$, as $f$ ranges over the set 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 $(1-α )/k$.