# MathSciDoc: An Archive for Mathematician ∫

#### Functional Analysismathscidoc:1701.12010

Arkiv for Matematik, 1-17, 2016.2
Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .
@inproceedings{fredrik2016fourier,
title={Fourier dimension of random images},
author={Fredrik Ekström},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203408147069826},
booktitle={Arkiv for Matematik},
pages={1-17},
year={2016},
}

Fredrik Ekström. Fourier dimension of random images. 2016. In Arkiv for Matematik. pp.1-17. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203408147069826.