Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. (English) Zbl 1408.11083

Bourgain, Jean; Demeter, Ciprian; Guth, Larry

Ann. Math. (2) 184, No. 2, 633-682 (2016).

This is a most important paper, proving essentially optimal exponents for the estimation of Vinogradov’s mean value.
Let \(s,n\) and \(N\) be positive integers and let \(J_{s,n}(N)\) be the Vinogradov mean value \[ \int_0^1\ldots\int_0^1 \left|\sum_{j\le N}e(x_1j+\ldots+x_nj^n)\right|^{2s}\,dx_1\ldots dx_n. \] Then it is proved that \[ J_{s,n}(N)\ll_{s,n,\varepsilon}N^{s+\varepsilon}+N^{2s-n(n+1)/2+\varepsilon} \] for any \(\varepsilon>0\). This estimate is optimal, apart from a possible factor \(N^{\varepsilon}\), which can be removed when \(s>n(n+1)/2\). It would be good to be able to prove such a bound with a strong dependence on \(s\) and \(n\).
The cases \(s=1\) and \(s=2\) are completely elementary, while the case \(s=3\) was recently resolved by T. D. Wooley [Adv. Math. 294, 532–561 (2016; Zbl 1365.11097)].
In contrast to previous approaches to the problem, the current paper uses techniques from harmonic analysis, and in particular builds on the recent developments in decoupling theory, see [J. Bourgain and C. Demeter, Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014)], for example. Also crucial are results on multilinear Kakeya problems, as in the work of J. Bennett et al. [Acta Math. 196, No. 2, 261–302 (2006; Zbl 1203.42019)], and an “induction on scales” method, taken from the work of J. Bourgain and C. Demeter [J. Anal. Math. 133, 279–311 (2017; Zbl 1384.42016)].

Reviewer: D. R. Heath-Brown (Garsington)

MSC:

11L07	Estimates on exponential sums
42B37	Harmonic analysis and PDEs

Keywords:

exponential sum; mean value; Vinogradov’s mean value; optimal exponent; decoupling; curves

Citations:

Zbl 1365.11097; Zbl 1322.42014; Zbl 1203.42019; Zbl 1384.42016

References:

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