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

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)].


11L07 Estimates on exponential sums
42B37 Harmonic analysis and PDEs
Full Text: DOI arXiv


[1] J. Bennett, A. Carbery, and T. Tao, ”On the multilinear restriction and Kakeya conjectures,” Acta Math., vol. 196, iss. 2, pp. 261-302, 2006. · Zbl 1203.42019
[2] J. Bennett, A. Carbery, M. Christ, and T. Tao, ”Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities,” Math. Res. Lett., vol. 17, iss. 4, pp. 647-666, 2010. · Zbl 1247.26029
[3] J. Bennett, N. Bez, T. Flock, and S. Lee, Stability of Brascamp-Lieb constant and applications.
[4] J. Bourgain, ”Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces,” Israel J. Math., vol. 193, iss. 1, pp. 441-458, 2013. · Zbl 1271.42039
[5] J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, 2014. · Zbl 1352.11065
[6] J. Bourgain, Decoupling inequalities and some mean-value theorems, 2014. · Zbl 1380.42008
[7] J. Bourgain and C. Demeter, ”The proof of the \(l^2\) Decoupling Conjecture,” Ann. of Math., vol. 182, iss. 1, pp. 351-389, 2015. · Zbl 1322.42014
[8] J. Bourgain and C. Demeter, Mean value estimates for Weyl sums in two dimensions, 2015. · Zbl 1408.11082
[9] J. Bourgain and C. Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, 2014. · Zbl 1384.42016
[10] J. Bourgain and C. Demeter, ”Decouplings for surfaces in \(\mathbbR^4\),” J. Funct. Anal., vol. 270, iss. 4, p. 1299–1318, 2016. · Zbl 1332.53007
[11] J. Bourgain and N. Watt, Decoupling for perturbed cones and mean square of \(\zeta(\frac12+it)\), 2015.
[12] J. Bourgain and L. Guth, ”Bounds on oscillatory integral operators based on multilinear estimates,” Geom. Funct. Anal., vol. 21, iss. 6, pp. 1239-1295, 2011. · Zbl 1237.42010
[13] K. Ford and T. D. Wooley, ”On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing,” Acta Math., vol. 213, iss. 2, pp. 199-236, 2014. · Zbl 1307.11102
[14] G. Garrigós and A. Seeger, ”On plate decompositions of cone multipliers,” Proc. Edinb. Math. Soc., vol. 52, iss. 3, pp. 631-651, 2009. · Zbl 1196.42010
[15] L. Guth, ”A short proof of the multilinear Kakeya inequality,” Math. Proc. Cambridge Philos. Soc., vol. 158, iss. 1, pp. 147-153, 2015. · Zbl 1371.42007
[16] M. Pramanik and A. Seeger, ”\(L^p\) regularity of averages over curves and bounds for associated maximal operators,” Amer. J. Math., vol. 129, iss. 1, pp. 61-103, 2007. · Zbl 1161.42009
[17] T. Wolff, ”Local smoothing type estimates on \(L^p\) for large \(p\),” Geom. Funct. Anal., vol. 10, iss. 5, pp. 1237-1288, 2000. · Zbl 0972.42005
[18] T. D. Wooley, ”The cubic case of the main conjecture in Vinogradov’s mean value theorem,” Adv. Math., vol. 294, pp. 532-561, 2016. · Zbl 1365.11097
[19] T. D. Wooley, Approximating the main conjecture in Vinogradov’s mean value Theorem, 2014. · Zbl 1372.11086
[20] T. D. Wooley, Translation invariance, exponential sums, and Waring’s problem. · Zbl 1373.11061
[21] T. D. Wooley, ”The asymptotic formula in Waring’s problem,” Int. Math. Res. Not., vol. 2012, iss. 7, pp. 1485-1504, 2012. · Zbl 1267.11104
[22] T. D. Wooley, Discrete Fourier restriction via efficient congruencing. · Zbl 1267.11105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.