  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
$N^{\epsilon}$.
