We prove the optimal L4-Strichartz estimate for the Schrödinger equation
on the two-dimensional rational torus, which improves an estimate of Bourgain. Instead of analytic number theory we employ the Szemeredi–Trotter Theorem, which gives an upper bound on the number of lines m passing through at least k points of a given set of n points in the plane. This new approach yields an even stronger L4 bound on a logarithmic time scale, which implies
global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger
equation in Hs for any s > 0 and data which is small in the critical L2 norm. This is joint work with Beomjong Kwak (KAIST).
