An $L^4$ maximal estimate for quadratic Weyl sums

Abstract: We show that $$\bigg|\sup_{0 < t < 1} \big|\sum_{n=1}^{N} e^{2\pi i (n(\cdot) + n^2 t)}\big| \bigg|{L^{4}([0,1])} \leq C{\epsilon} N^{3/4 + \epsilon}$$ and discuss some applications to the theory of large values of Weyl sums. This estimate is sharp for quadratic Weyl sums, up to the loss of $N^{\epsilon}$.