Abstract
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 \(m\).
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Ekström, F. Fourier dimension of random images. Ark Mat 54, 455–471 (2016). https://doi.org/10.1007/s11512-016-0237-3
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DOI: https://doi.org/10.1007/s11512-016-0237-3