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On the Vinogradov mean value. (English, Russian) Zbl 1371.11138
Proc. Steklov Inst. Math. 296, 30-40 (2017); translation in Tr. Mat. Inst. Steklova 296, 36-46 (2017).
Writing \(e(t)=e^{2\pi it}\), for positive integers \(k,s\) and \(x\in{\mathbb R}^k\) let \[ f_k(x,N)=\sum_{1\leq n\leq N} e(nx_1+n^2x_2+\ldots+n^kx_k) \] and \[ J_{s,k}(N)=\int_{[0,1]^k} |f_k(x,N)|^{2s}\,dx_1\cdots dx_k. \] As it is well-known, \(J_{s,k}\) counts the number of integer solutions of the system \(n_1^j+\ldots+n_s^j=n_{s+1}^j+\ldots+n_{2s}^j\) \((1\leq j\leq k)\), where \(1\leq n_i\leq N\) \((i=1,\dots,2s)\). The evaluation of \(J_{s,k}\) is a central problem, with many important applications. The so-called Main Conjecture (going back to Vinogradov) states that \(|J_{s,k}|\ll N^\varepsilon(N^s+N^{2s-k(k+1)/2})\) for all \(\varepsilon>0\). In the paper a discussion is given on a recent work of C. Demeter, L. Guth and the author [Ann. Math. (2) 184, No. 2, 633–682 (2016; Zbl 1408.11083)] where they proved the Main Conjecture using the decoupling theory for curves.

MSC:
11P55 Applications of the Hardy-Littlewood method
11L15 Weyl sums
11L07 Estimates on exponential sums
11P05 Waring’s problem and variants
Citations:
Zbl 1408.11083
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References:
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