Obtaining the Fourier spectrum via Fourier coefficients

The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and is defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete representation of those energies using Fourier coefficients. As an example application, we use this representation to establish sharp bounds for the Fourier spectrum of a general measure with bounded support, improving previous estimates of the second-named author.