The study of maximal estimates for Weyl sums has been an active subject in harmonic analysis. It links to a problem of Carleson that asks for the smallest number s for which the solution to the periodic Schrodinger equation converges pointwise back to the initial datum which lies in the Sobolev space of order s. The question of how small the number s boils down to that of how well we can get the maximal estimate. A current conjecture of maximal estimate suggests that, if true, s should be no lower than a quarter. In this talk, we will introduce a result of Barron where he shows the maximal estimate conjecture holds for a specific form of Weyl sums.