TY - JOUR AU - Armstrong, Gavin AU - O'Farrell, Anthony G. PY - 2022/02/24 Y2 - 2024/03/28 TI - Dimension of images of large level sets JF - MATHEMATICA SCANDINAVICA JA - Math. Scand. VL - 128 IS - 1 SE - Articles DO - 10.7146/math.scand.a-129246 UR - https://www.mscand.dk/article/view/129246 SP - AB - <p>Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\colon E\to \mathbb{R} $, where $E$ is some interval on the line having positive length. For $0&lt;\alpha &lt;1$ let $I_\alpha (f)$ denote the set of values $y\in \mathbb{R} $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\alpha$. We consider how large can be the Hausdorff dimension of $I_\alpha (f)$, 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 $\dim I_\alpha (f)$ is $(1-\alpha )/k$.</p> ER -