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 ).
The Fourier dimension is not, in general, stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.