An extended Vinogradov's mean value theorem

TL;DR

This paper advances the Vinogradov's mean value theorem by establishing new mean value estimates for exponential sums using the Hardy-Littlewood circle method and refined shifting variables, achieving sharp bounds for certain dimensions.

Contribution

It introduces novel mean value bounds for exponential sums related to Vinogradov's conjecture, employing refined analytical techniques and decoupling inequalities for higher dimensions.

Findings

Sharp upper bounds for mean values when d=2,3.

Results depend on decoupling inequalities for d≥4.

Enhanced understanding of exponential sums in number theory.

Abstract

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov's mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument. Let $d\geq 2$ be a natural number and $\boldsymbol{\alpha}=(\alpha_d,\ldots, \alpha_1)\in \mathbb{R}^d.$ Define the exponential sum \begin{equation*} f_d(\boldsymbol{\alpha};N):=\sum_{1 \leq n \leq N}e(\alpha_d n^d + \cdots+ \alpha_1 n). \end{equation*} For $p>0$, consider mean values of the exponential sums \begin{equation*} \mathcal{I}_{p,d}(u;N):=\int_{[0,1)\times [0,N^{-u})\times [0,1)^{d-2}}|f_d(\boldsymbol{\alpha};N)|^pd\boldsymbol{\alpha}, \end{equation*} where we wrote $d\boldsymbol{\alpha}=d\alpha_1 d\alpha_2\cdots d\alpha_{d-1}d\alpha_d.$ By making use of the aforementioned tools, we obtain the sharp upper bound for $\mathcal{I}_{p,d}(u;N)$, for $d=2,3$ and $0<u\leq 1$. Furthermore, for $d \geq 4$, we obtain analogous results depending on a small cap decoupling inequality for the moment curves in $\mathbb{R}^d.$