Research

I'm interested in how tools from Fourier analysis can reveal information about how structured/random sets are. 'Traditional' notions of dimension describe how fractal sets are distributed in space, but the Fourier dimension gives us information about how structured the sets are, from an arithmetical point of view. This perspective connects with many classical themes in fractal geometry, such as orthogonal projections, and distance sets.
At the same time, fractal sets are themselves a natural space in which we can revisit questions from harmonic analysis. Many problems change their character when posed on fractal measures rather than on smooth ones. This shift often brings new phenomena into light. I'm interested in exploring how ideas from dimension theory can shed light on restriction-type problems for fractal measures.

Preprints

3. $L^2$ restriction estimates from the Fourier spectrum (with M. Carnovale and J. M. Fraser), arXiv.
   Submitted (2024)
   

To appear

1. Obtaining the Fourier spectrum via Fourier coefficients (with M. Carnovale and J. M. Fraser), arXiv.
   Proceedings of the American Mathematical Society (to appear)
   
2. A Fourier analytic approach to exceptional set estimates for orthogonal projections (with J. M. Fraser), arXiv.
   Indiana University Mathematics Journal (to appear)