Improved global well-posedness for mass-critical nonlinear Schr\"odinger equations on tori

We show new global well-posedness results for mass-critical nonlinear Schr\"odinger equations on tori in one and two dimensions. For the quintic nonlinear Schr\"odinger equation on the circle we show global well-posedness for initial data in $H^s(\mathbb{T})$ for $s>\frac{1}{3}$ and $\| u_0 \|_{L^2(\mathbb{T})} \ll 1$. In two dimensions we show global well-posedness on possibly irrational tori for $s>\frac{3}{5}$. In the focusing case we need a smallness condition for the mass, whereas in the defocusing case large mass is covered.