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Vinogradov’s mean value theorem via efficient congruencing. II. (English) Zbl 1312.11066
Let \(e(z)=\mathrm{e}^{2\pi i z}\), and \(k,s,r,t\in\mathbb{N}\). Also let \(\alpha=(\alpha_1,\dots,\alpha_k)\in\mathbb{R}^k\), and define \[ f_k(\alpha;X)=\sum_{1\leq x\leq X}e(\alpha_1 x+\dots+\alpha_k x^k), \] The main purpose of the present impressive paper is to estimate the Vinogradov’s integral defined by \[ J_{s,k}(X)=\int_{[0,1)^k}\left|f_k(\alpha;X)\right|^{2s}\mathrm{d}\alpha. \] The author obtains several estimates regarding the values of \(s\). Gathering his main results, he proves that for each \(\varepsilon>0\) one has \[ J_{s,k}(X)\ll \begin{cases} X^{2s-\frac12 k(k+1)+\varepsilon} & \text{for \(k\geq 3\) and \(s\geq k^2-1\)},\\ X^{s+\frac{r-1}{k-r}+\varepsilon} & \text{for \(k\geq 3\), \(1\leq r\leq \kappa_{\min}\) and \(s\leq \kappa_r\)},\\ X^{2s-\frac12 k(k+1)+\Delta_{t,k}+\varepsilon} & \text{for \(k\geq 3\), \(1\leq t\leq k-1\) and \(s\geq \kappa_t\)}, \end{cases} \] where \(\kappa_{\min}=\min\{k-2,\frac12k+1\}\), \(\kappa_r=r(k-r+2)\), \(\kappa_t=(k-t)(k+1)\) and \(\Delta_{t,k}=\frac12t(t-1)(\frac{k+1}{k-1})\). As a corollary of the mid mentioned bound, he obtains the estimate \(J_{s,k}(X)\ll X^{s+1+\varepsilon}\) for \(k\geq 4\) and \(s\leq \frac14k^2+k\), and for each \(\varepsilon>0\).
Several applications of these kind of estimates are given in the paper, including applications in the context of Waring’s problem, improvements analogues of Weyl’s inequality, the distribution of polynomials modulo 1, Tarry’s problem on the Diophantine systems, and improvement in a result of E. Croot and D. Hart [SIAM J. Discrete Math. 24, No. 2, 505–519 (2010; Zbl 1221.11202)] related to the sum-product theorem. The paper under review is a continuation of the author’s recent work [Ann. Math. (2) 175, No. 3, 1575–1627 (2012; Zbl 1267.11105)].

MSC:
11L15 Weyl sums
11L07 Estimates on exponential sums
11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
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