Abstract: The Bochner--Riesz problem is closely related to the convergence of multi-dimensional Fourier series. The classical version of this problem is fully resolved in the plane, where one considers partial sums of the Fourier series taken over discs of radius N centred at the origin, and studies their convergence as N tends to infinity. Seeger and Ziesler investigated a variant of this problem in the plane, by replacing discs with more general convex domains, and found that the convergence behaviour depends on a fractal analytic quantity known as the affine Minkowski dimension of the domain.
In this talk, we will give a soft introduction to the fractal Bochner--Riesz problem, discussing the notion of the affine Minkowski dimension, and study the problem using newly developed fractal decoupling theorems, both by myself and by others.
