Arkiv för Matematik

Volume 61 (2023)

Number 2

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

Pages: 267 – 276



Julia Brandes (Mathematical Sciences, University of Gothenburg and Chalmers Institute of Technology, Göteborg, Sweden)

Igor E. Shparlinski (Department of Pure Mathematics, University of New South Wales, Sydney, NSW, Australia)


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.


exponential sums

2010 Mathematics Subject Classification

Primary 11L15. Secondary 11J83, 11T23.

During the preparation of this manuscript, JB was supported by Starting Grant no. 2017-05110 of the Swedish Science Foundation (Vetenskapsrådet) and IS was supported by the Australian Research Council Grant DP170100786.

Received 8 February 2022

Received revised 17 November 2022

Accepted 13 December 2022

Published 13 November 2023