We consider the Bochner--Riesz means associated with planar convex domains introduced by Seeger--Ziesler. Using ideas from additive combinatorics, we construct domains for which the associated Bochner--Riesz means are bounded on a wider range of exponents compared to the universal range obtained by Seeger--Ziesler.

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form x, x+y, x+y^2. We discuss a multidimensional analog of this result, previously proved using dynamical methods. In particular, we show how harmonic analysis can yield quantitative bounds on the size of sets lacking such configurations. This is joint work with Sarah Peluse and Fernando Xuancheng Shao.

Let r_1(n) be  the number of integers up to x that can be written as the square of an integer plus the square of a prime. We discuss the erratic behaviour of r_1, which is similar to the one of the divisor function. As well we will show that its mean is asymptotic to (pi/2) x log x minus a secondary term of size x/(logx)^{1+d+o(1)}, where d is the multiplication table constant. Detailed heuristics suggest very precise asymptotic for the secondary term as well. In particular, our proofs imply that the main contribution to the mean value of r_1(n) comes from integers with “unusual” number of prime factors, i.e, those with omega(n) ~ 2 log log x (for which r_1(n) ~ (log x)^{log 4-1}), where omega(n) is the number of district prime factors of n. 

In the talk we will review the results of two works: my paper from 2022 and a recent joint preprint with Andrew Granville and Cihan Sabuncu.

The massless Dirac equation with a Coulomb potential is interesting both from a physical and a mathematical point of view; it appears in some physical models, for instance the 2D equation is used to describe the dynamics of carbon atoms in a sheet of non-perfect graphene, and on the mathematical side the homogeneity of degree -1 of the potential seems to have a critical behavior, as |x| goes to infinity, since Strichartz estimates are known to hold for potentials that decay faster and there are examples of potentials decaying slower such that the corresponding flows do not disperse.

In this talk I will present a recent result concerning Strichartz estimates for the solutions of the massless Dirac-Coulomb equation in 2 and 3 dimension with additional angular regularity. It extends the result on R^3 of Cacciafesta-Séré-Zhang and provides completely new estimates on R^2.

As an application we will discuss a local well-posedness result for a nonlinear system.

Pdf version.