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 $s<d\mathrm{/}(d+2)$, in all dimensions $d\ge 2$. We construct normalized data with frequencies of size $N$ whose evolution attains size $N^{d\mathrm{/}(d+2)}$ on a set of uniformly positive measure. The key ingredient in the construction is that the frequencies lie in an affine congruence class, so that at suitable rational times their phases coincide on a family of well-separated spatial points. The result matches the known estimate for $s>d\mathrm{/}(d+2)$ with $d\ge 2$. By integer dilation and uniform boundedness, we also obtain a single datum in $H^s$ whose evolution is unbounded along a sequence of times tending to zero whenever $s<d\mathrm{/}(d+2)$. We also record a logarithmic upper bound in $2D$ at the critical frequency power and a lower bound on shrinking time intervals.