This survey shows how the Assouad spectrum, intermediate dimensions, and Fourier spectrum yield sharper projection theorems for fractal sets.
Obtaining the Fourier spectrum via Fourier coefficients
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abstract
The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete representation of those energies using the 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
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Applications of dimension interpolation to orthogonal projections
This survey shows how the Assouad spectrum, intermediate dimensions, and Fourier spectrum yield sharper projection theorems for fractal sets.