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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
Zbl 1408.11083
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