On the distribution of the maximum of cubic exponential sums

TL;DR

This paper studies the distribution of the maximum partial sums of cubic exponential sums, providing bounds that improve previous results and applying probabilistic, harmonic analysis, and algebraic geometry techniques.

Contribution

It offers nearly tight bounds for the distribution of large values of cubic exponential sums, extending results to Kloosterman sums and general $ ext{l}$-adic trace functions.

Findings

Bounds on the maximum of cubic exponential sums are nearly optimal.

The results apply uniformly across a wide range of parameters.

Existence of large partial sums of exponential sums with specific bounds.

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.