On the distribution of the maximum of cubic exponential sums

Abstract

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1, p]$ and $a\in \mathbb{F}_p^{\times}$ such that $|\sum_{n\leq x} \exp(2\pi i (n^3+an)/p)|\ge (2/\pi+o(1)) \sqrt{p}\log\log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.