Two-dimensional Weyl sums failing square-root cancellation along lines

TL;DR

This paper demonstrates that certain two-dimensional Weyl sums can attain large values along linear slices, challenging the expectation of square-root cancellation and extending previous results to general polynomials.

Contribution

It extends the understanding of Weyl sums' behavior from quadratic and cubic cases to arbitrary degree polynomials, revealing large value phenomena along lines.

Findings

Weyl sums can reach sizes of P^{3/4 + o(1)} on almost all linear slices.

Contradicts the expectation of square-root cancellation for generic subvarieties.

Extends prior results from quadratic and cubic to general polynomial degrees.

Abstract

We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.