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An extended Vinogradov's mean value theorem (2506.01751v2)

Published 2 Jun 2025 in math.NT

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 nd + \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 $\mathbb{R}d.$

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