The convergence problem for the Schr\”odinger equation, one of the most famous problems in Harmonic Analysis, asks for the smallest Sobolev regularity $s$ such that $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ for almost every $x$ and for all $f \in H^s$. Recent works have shown that the optimal regularity is $s =n/(2(n+1))$. However, there are several relevant cases in which the problem is still open, such as the periodic equation, the equation with other symbols like the fractional laplacian $\Delta^\sigma$, and the fractal convergence $\mathcal H^\alpha$-almost everywhere, where $\mathcal H^\alpha$ are Hausdorff measures.

In this talk I will discuss how to exploit Bourgain’s optimal counterexample to improve the results for these unsolved problems, using for that the Mass Transference Principle, a technique from Diophantine approximation related to the Duffin-Schaeffer conjecture. I will devote the first part of the talk to introduce the problem, the classical results and the main techniques used, as well as to dissect Bourgain’s counterexample for the classical problem.

This is based on a collaboration with Felipe Ponce-Vanegas (BCAM, Spain).