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Quantitative flatness and obstructions in Fourier analysis
Jonathan M. Fraser
Three important problems in Fourier analysis are the Fourier restriction problem, the -improving problem, and the Fourier decay problem. Positive results for any of these prob…
Fourier restriction estimates based on -dimensions: beyond Stein--Tomas
Marc Carnovale, Jonathan M. Fraser, Ana E. de Orellana
The well-known Stein--Tomas restriction theorem gives the sharp range of for which restriction estimates hold for the surface measure on the sphere. This was gener…
Fourier analytic variants of the Furstenberg and Kakeya problems
Jonathan M. Fraser, Lijian Yang
We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given , we call a set $K \…
On Fourier decay and the distance set problem
Jonathan M. Fraser, Thang Pham
We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dime…
An invitation to dimension interpolation
Jonathan M. Fraser
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill u…
restriction estimates from the Fourier spectrum
Marc Carnovale, Jonathan M. Fraser, Ana E. de Orellana
The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of for which restriction estimates hold for a given mea…
Fourier decay of product measures
Jonathan M. Fraser
Can one characterise the Fourier decay of a product measure in terms of the Fourier decay of its marginals? We make inroads on this question by describing the Fourier spectrum of a…
A Fourier analytic approach to exceptional set estimates for orthogonal projections
Jonathan M. Fraser, Ana E. de Orellana
Marstrand's celebrated projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. It is…
Obtaining the Fourier spectrum via Fourier coefficients
Marc Carnovale, Jonathan M. Fraser, Ana E. de Orellana
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 dec…