Dimension of images of large level sets

Authors

Gavin Armstrong, Central Washington UniversityFollow
Anthony O'Farrell, Maynooth University

Document Type

Article

Department or Administrative Unit

Mathematics

Publication Date

2-24-2022

Abstract

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 I_α(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 dimI_α(f) is (1−α)/k.

Comments

This article was originally published in Mathematica Scandinavica. The full-text article from the publisher can be found here.

Due to copyright restrictions, this article is not available for free download from ScholarWorks @ CWU.

Recommended Citation

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

Journal

Mathematica Scandinavica