Dimension of images of large level sets
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Let k be a natural number. We consider k-times continuously differentiable real-valued functions f:E→R, where E is some interval on the line having positive length. For 0<α<1 let Iα(f) denote the set of values y∈R whose preimage f−1(y) has Hausdorff dimension at least α. We consider how large can be the Hausdorff dimension of Iα(f), as f ranges over the set of all k-times continuously differentiable functions from E into R. We show that the sharp upper bound on dimIα(f) is (1−α)/k.
Armstrong, G., & O'Farrell A. G. (2022). Dimension of images of large level sets. Mathematica Scandinavica, 128(1), 147-160. https://doi.org/10.7146/math.scand.a-129246
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