An L4 Maximal Estimate for Quadratic Weyl Sums

Abstract

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 $N^{\epsilon }$.