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

2020/12/16 by Julia Brandes, Brandes, Julia, Igor E. Shparlinski +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.08877

openalex publication_date 2020/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• New bounds of Weyl sums
• On a Class of Sums with Unexpectedly High Cancellation, and its Applications
• On L¹² square root cancellation for exponential sums associated with nondegenerate curves in \mathbb R⁴
• On the cubic Weyl sum
• Exponential sums over primes are unbounded