A sharp regularity threshold for Schrödinger maximal estimates on standard tori

We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus \(\mathbb{T}^d\) for initial data \(f\in H^s(\mathbb{T}^d)\) when \(s<d/(d+2)\), in all dimensions \(d\ge2\). We construct normalized data with frequencies of size $N$ whose evolution attains size $N^{d/(d+2