[Paper Review] A fourth derivative test for exponential sums

Robert Osman, Patrick Sargos|arXiv (Cornell University)|Jul 7, 2023

Analytic Number Theory Research4 citations

TL;DR

This paper improves the classical van der Corput fourth derivative exponential sum bound from λ¹/¹⁴ to λ¹/¹³ by reducing the problem to a mean square estimate for triple exponential sums, leveraging Bombieri and Iwaniec's double large sieve and a novel application of Taylor expansion and partial summation. The key result is a sharp Oε(M¹⁺ελ¹/¹³) bound under M ≫ λ⁻⁸/¹³.

ABSTRACT

We give an upper bound for the exponential $\sum_{m=1}^M \exp( 2iπf (m))$ in terms of $M$ and $λ$, where $λ$ is a small positive number which denotes the size of the fourth derivative of the real valued function $f$. The classical van der Corput's exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums.

Motivation & Objective

• To improve the classical van der Corput bound for exponential sums with fourth derivative control.
• To achieve a better exponent than 1/14 without stronger hypotheses on the phase function.
• To extend the range of applicability to shorter exponential sums via a refined analysis of diophantine systems.
• To provide a self-contained, elementary proof relying on A-processes and Taylor expansion.
• To establish a new exponent pair (1/13 + ε, 12/13 + ε) for the class of semi-monomial functions.

Proposed method

• Apply the van der Corput A-process twice to transform the original sum into a quadruple exponential sum.
• Use a shift of the main variable to reorganize the sum into a form amenable to double large sieve techniques.
• Expand the phase function using Taylor's formula up to the fourth derivative, isolating the error term.
• Apply partial summation to control the error term arising from the remainder in Taylor expansion.
• Reduce the problem to counting solutions of a specific Diophantine system involving second and third derivatives.
• Use Bombieri and Iwaniec's double large sieve to bound the number of solutions, leading to the improved exponent.

Experimental results

Research questions

• RQ1 Can the classical van der Corput exponent 1/14 for fourth derivative exponential sums be improved under the same hypotheses?
• RQ2 What is the optimal exponent ϑ such that SM ≪ε M¹⁺ελϑ holds under the fourth derivative condition f⁽⁴⁾(x) ≍ λ?
• RQ3 Can the bound be extended to shorter sums with M ≪ λ⁻⁸/¹³ while maintaining a non-trivial error term?
• RQ4 Is the exponent 1/13 sharp, or can it be improved further under the same assumptions?
• RQ5 What is the minimal β such that SM ≪ Mλ¹/¹⁴ holds for M ≫ λ⁻β under the fourth derivative condition?

Key findings

• The paper establishes the bound SM ≪ε M¹⁺ελ¹/¹³ for exponential sums under the condition M ≫ λ⁻⁸/¹³.
• An equivalent bound is SM ≪ε Mε(Mλ¹/¹³ + λ⁻⁷/¹³) is proven, which is effective even when M is small.
• The improvement from 1/14 to 1/13 is achieved via a novel reduction to a mean square estimate for triple exponential sums.
• The proof relies on a double large sieve applied to a Diophantine system derived from Taylor expansion of the phase function.
• The method is self-contained and elementary, avoiding deep tools beyond classical exponential sum techniques.
• The result implies that (1/13 + ε, 12/13 + ε) is an exponent pair for semi-monomial functions, improving upon prior bounds under the same conditions.