Given a compact set of real numbers, a random (C^ m + )-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 + ) ). This is used to show that every Borel subset of the real numbers of Hausdorff dimension (s ) is (C^ m + )-equivalent to a set of Fourier dimension greater than or equal to (s (m + ) ). 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 ).
In @cite_4 , Bluhm gave a method for randomly perturbing a class of self-similar measures on @math , such that the perturbed measure almost surely has Fourier dimension equal to the similarity dimension of the original measure. For @math , the uniform measures on Cantor sets with constant contraction ratio are among the measures considered by Bluhm, and if the parameters in the construction are chosen suitably then the perturbation is a bi-Lipschitz map. Thus it follows from Bluhm's result that such Cantor sets are bi-Lipschitz equivalent to Salem sets.
@cite_4
2021822733
In this paper we investigate the pointwise Fourier decay of some selfsimilar random measures. As an application we construct statistically selfsimilar Salem sets. For example, our result shows that a “slight” random perturbation of the classical Cantor set becomes a “nice” set in the sense that its Fourier dimension equals its Hausdorff dimension.