Bilinear Forms with Exponential Sums with Binomials

TL;DR

This paper investigates bilinear exponential sums involving binomials, establishing estimates and demonstrating nontrivial cancellations when coefficients vary over sparse sets, advancing understanding of exponential sum behavior.

Contribution

It provides new estimates for bilinear exponential sums with binomials and shows cancellations over sparse coefficient sets, a novel contribution in this area.

Findings

Established bounds for bilinear exponential sums with binomials.

Proved existence of nontrivial cancellations in sums over sparse coefficient sets.

Enhanced understanding of exponential sum behavior with binomial phases.

Abstract

We obtain several estimates for bilinear form with exponential sums with binomials $mx^k + nx^\ell$. In particular we show the existence of nontrivial cancellations between such sums when the coefficients $m$ and $n$ vary over rather sparse sets of general nature