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Bounds for the Quartic Weyl Sum

D. R. Heath-Brown

Abstract

We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that \[\sum_{n\le N} e(αn^4)\ll_{\ep,α}N^{5/6+\ep}\] for any $\ep>0$ and any quadratic irrational $α\in\R-\Q$. Classically one would have had the exponent $7/8+\ep$ for such $α$. In contrast to the author's earlier work \cite{cubweyl} on cubic Weyl sums (which was conditional on the $abc$-conjecture), we show that the van der Corput $AB$-steps are sufficient for the quartic case, rather than the $BAAB$-process needed for the cubic sum.

Bounds for the Quartic Weyl Sum

Abstract

We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that for any and any quadratic irrational . Classically one would have had the exponent for such . In contrast to the author's earlier work \cite{cubweyl} on cubic Weyl sums (which was conditional on the -conjecture), we show that the van der Corput -steps are sufficient for the quartic case, rather than the -process needed for the cubic sum.
Paper Structure (5 sections, 5 theorems, 40 equations)

This paper contains 5 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. The van der corput $A$-Process
  3. The van der corput $B$-Process
  4. Deduction of Theorem \ref{['sqrt2']}
  5. Addendum

Key Result

Theorem 1

Let $\alpha\in\mathbb{R}-\mathbb{Q}$ be a quadratic irrational. Then for any $\varepsilon>0$.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['genthm']}
  • Theorem 3