Bounds for the Cubic Weyl Sum

TL;DR

This paper establishes a conditional upper bound on cubic Weyl sums, demonstrating that their growth is limited by a power of N under the abc-conjecture for quadratic irrational lpha.

Contribution

It provides a new conditional bound on cubic Weyl sums assuming the abc-conjecture, linking deep conjectures in number theory to exponential sum estimates.

Findings

Conditional bound or Weyl sums: or N^{5/7+psilon}

Results depend on the validity of the abc-conjecture

Advances understanding of exponential sums involving cubic polynomials.

Abstract

It is shown, subject to the abc-conjecture, that \[\sum_{n\le N}\exp(2\pi i\alpha n^3)\ll_{\epsilon,\alpha}N^{5/7+\epsilon}\] for any $\epsilon>0$ and any quadratic irrational $\alpha$.