arXiv 1802.097012018-02-27math.NT

关于三次指数和最大值的分布

On the distribution of the maximum of cubic exponential sums

Youness Lamzouri

备注：: 28 pages
发表信息：: J. Inst. Math. Jussieu 19 (2020) 1259-1286
DOI：: 10.1017/S1474748018000385

摘要：

本文研究了一类三次指数和的部分和的最大值的分布。我们的主要定理给出了这个最大值的大值分布的上界和下界（几乎相同的数量级），它们在很宽的均匀范围内成立。这改进了Kowalski和Sawin最近的一个结果。证明混合使用概率方法，调和分析技术，和代数几何的深层工具。所得结果也可推广到其它类型的$\ell$ -adic迹函数。特别地，我们的结果的下界也适用于Kloosterman和的部分和。作为应用，我们证明了存在$x\in [1, p]$和$a\in \mathbb{F}_p^{\times}$使得$|\sum_{n\leq x} \exp(2πi (n^3+an)/p)|\ge (2/π+o(1)) \sqrt{p}\log\log p$ .我们的结果的一致性表明，这个界限是最佳的，常数的值。

英文摘要：

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 $|\sum_{n\leq x} \exp(2πi (n^3+an)/p)|\ge (2/π+o(1)) \sqrt{p}\log\log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

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