Igor Shparlinski: Weyl Sums: Large, Small and Typical (NTWS 015)
Channel: Number Theory Web Seminar
Date: 2023-04-12

Abstract: While Vinogradov’s Mean Value Theorem, in the form given by J. Bourgain, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019), gives an essentially optimal result on the power moments  of the Weyl sums 
S(u;N) =\sum_{1\le n \le N} \exp(2 \pi i (u_1n+…+u_dn^d))
where u = (u_1,...,u_d) \in [0,1)^d,  very little is known about the distribution, or even existence, of u \in [0,1)^d, for which these sums are very large, or small, or close to their average value N^{1/2}. In this talk, we describe recent progress towards these and some related questions.
We also present some new bounds on S(u;N) which hold for almost all (u_i)_{i\in I} and all (u_j)_{j\in J}, where I \cup J is a partition of {1,…,,d}. These bounds improve similar results of T. Wooley (2015). Our method also applies to binomial sums 
T(x,y; N) = \sum_{1\le n \le N} \exp(2 \pi i (xn+yn^d))
with x,y \in [0,1), in which case we improve some results of M.B. Erdogan and G. Shakan (2019).
This is a joint work with Changhao Chen and Bryce Kerr.

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Tuesday, June 23, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)