Igor Shparlinski: Weyl Sums: Large, Small and Typical

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) =∑_{1 ≤ n ≤ N} exp(2πi (u₁n+ . . . +u_dn^d)),

where u = (u₁,…,u_d) ∈ [0,1)^d, very little is known about the distribution, or even existence, of u ∈ [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∈I and all (u_j)_j∈J, where I ∪ 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) = ∑_{1 ≤ n ≤ N} exp(2πi (xn+yn^d)) with x,y∈[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.

This video is part of the Number Theory Web Seminar series.