Error term improvements for van der Corput transforms

TL;DR

This paper enhances the error term analysis in the van der Corput transform for exponential sums, enabling more precise asymptotics and applicability to longer intervals without using the truncated Poisson formula.

Contribution

It introduces improved error bounds and extracts the next asymptotic term for a broad class of functions, surpassing previous methods like Karatsuba and Korolev.

Findings

Sharper error bounds for exponential sums

Ability to handle longer intervals more effectively

Validation of sharpness of previous results

Abstract

We improve the error term in the van der Corput transform for exponential sums \sum_{a \le n \le b} g(n) exp(2\pi i f(n)). For many functions g and f, we can extract the next term in the asymptotic, showing that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a,b] with far better results. We provide a detailed analysis of the error term in the case g(x)=1 and f(x)=(x/3)^{3/2}.