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 \(m\).

中文翻译：

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随机图像的傅立叶维数

给定一个紧凑的实数集，构造一个随机\（C ^ {m + \ alpha} \）-亚纯，使得任何量度的图像都集中在该集合上，并满足涉及实数\（s \ ），几乎可以确定其傅立叶尺寸大于或等于\（s /（m + \ alpha）\）。这用于表明Hausdorff维数\（s \）的实数的每个Borel子集都是\（C ^ {m + \ alpha} \）-等效于一组大于或等于\（s /（m + \ alpha）\）。特别地，每个Borel集对Salem集都是微分同构的，对于任何\（m \），在\（C ^ {m} \）-同构下傅立叶维数不是不变的。