Research

While the Hausdorff dimension of a set gives information regarding its geometric scaling properties, the Fourier dimension quentifies how structured vs how random that set is. For example, the Cantor set (very algebraically structured) has Fourier dimension zero, but random sets are usually Salem sets, i.e. their Fourier dimension is as large as it can be, being equal to their Hausdorff dimension.
In these situations, where we have two dimensions that capture different types of information, dimension interpolation is useful to gain insight on their relation. I'm currently studying the Fourier spectrum, a family of dimensions that live between the Fourier and the Hausdorff dimensions for sets and Fourier and Sobolev dimensions for measures.

Preprints

1. Obtaining the Fourier spectrum via Fourier coefficients (with M. Carnovale and J. M. Fraser), arXiv.
   Submitted (2024)
   + Abstract

To appear

1. A Fourier analytic approach to exceptional set estimates for orthogonal projections (with J. M. Fraser), arXiv.
   Indiana University Mathematics Journal (to appear)
   + Abstract