Two-dimensional Weyl sums failing square-root cancellation along lines
Number Theory
2020-12-17 v1
Abstract
We show that a certain two-dimensional family of Weyl sums of length takes values as large as 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.
Cite
@article{arxiv.2012.08877,
title = {Two-dimensional Weyl sums failing square-root cancellation along lines},
author = {Julia Brandes and Igor E. Shparlinski},
journal= {arXiv preprint arXiv:2012.08877},
year = {2020}
}