Proof of the main conjecture in Vinogradov’s Mean Value Theorem for degrees higher than three

Pages 633-682 from Volume 184 (2016), Issue 2 by Jean Bourgain, Ciprian Demeter, Larry Guth

Abstract

We prove the main conjecture in Vinogradov’s Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves.

  • [BCT] Go to document J. Bennett, A. Carbery, and T. Tao, "On the multilinear restriction and Kakeya conjectures," Acta Math., vol. 196, iss. 2, pp. 261-302, 2006.
    @ARTICLE{BCT, mrkey = {2275834},
      number = {2},
      issn = {0001-5962},
      author = {Bennett, Jonathan and Carbery, Anthony and Tao, Terence},
      mrclass = {42B20 (44A15 45P05 46E35)},
      doi = {10.1007/s11511-006-0006-4},
      journal = {Acta Math.},
      zblnumber = {1203.42019},
      volume = {196},
      mrnumber = {2275834},
      fjournal = {Acta Mathematica},
      mrreviewer = {Loukas Grafakos},
      coden = {ACMAA8},
      title = {On the multilinear restriction and {K}akeya conjectures},
      year = {2006},
      pages = {261--302},
      }
  • [BCCT] Go to document 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.
    @ARTICLE{BCCT, mrkey = {2661170},
      number = {4},
      issn = {1073-2780},
      author = {Bennett, Jonathan and Carbery, Anthony and Christ, Michael and Tao, Terence},
      mrclass = {26D15},
      doi = {10.4310/MRL.2010.v17.n4.a6},
      journal = {Math. Res. Lett.},
      zblnumber = {1247.26029},
      volume = {17},
      mrnumber = {2661170},
      fjournal = {Mathematical Research Letters},
      mrreviewer = {Pankaj Jain},
      title = {Finite bounds for {H}ölder-{B}rascamp-{L}ieb multilinear inequalities},
      year = {2010},
      pages = {647--666},
      }
  • [BBFL] J. Bennett, N. Bez, T. Flock, and S. Lee, Stability of Brascamp-Lieb constant and applications.
    @misc{BBFL,
      author = {Bennett, Jonathan and Bez, N. and Flock, T. and Lee, S.},
      TITLE={Stability of {Br}ascamp-{L}ieb constant and applications},
      arxiv = {1508.07502},
      }
  • [Bo2] Go to document J. Bourgain, "Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces," Israel J. Math., vol. 193, iss. 1, pp. 441-458, 2013.
    @ARTICLE{Bo2, mrkey = {3038558},
      number = {1},
      issn = {0021-2172},
      author = {Bourgain, Jean},
      mrclass = {42B08 (42B05)},
      doi = {10.1007/s11856-012-0077-1},
      journal = {Israel J. Math.},
      zblnumber = {1271.42039},
      volume = {193},
      mrnumber = {3038558},
      fjournal = {Israel Journal of Mathematics},
      mrreviewer = {F. M{ó}ricz},
      coden = {ISJMAP},
      title = {Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces},
      year = {2013},
      pages = {441--458},
      }
  • [Bo] J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, 2014.
    @misc{Bo,
      author = {Bourgain, Jean},
      TITLE = {Decoupling, exponential sums and the {R}iemann zeta function},
      arxiv = {1408.5794},
      YEAR={2014},
      }
  • [Bo6] J. Bourgain, Decoupling inequalities and some mean-value theorems, 2014.
    @misc{Bo6,
      author = {Bourgain, Jean},
      TITLE = {Decoupling inequalities and some mean-value theorems},
      NOTE={to appear in {\em J. d'Analyse Math.}},
      arxiv = {1406.7862},
      YEAR={2014},
      }
  • [BD3] Go to document J. Bourgain and C. Demeter, "The proof of the $l^2$ Decoupling Conjecture," Ann. of Math., vol. 182, iss. 1, pp. 351-389, 2015.
    @article {BD3, MRKEY = {3374964},
      AUTHOR = {Bourgain, Jean and Demeter, Ciprian},
      TITLE = {The proof of the {$l\sp 2$} {D}ecoupling {C}onjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {182},
      YEAR = {2015},
      NUMBER = {1},
      PAGES = {351--389},
      ISSN = {0003-486X},
      MRCLASS = {42B37 (11E76 46E30 53C40)},
      MRNUMBER = {3374964},
      MRREVIEWER = {G. V. Rozenblum},
      DOI = {10.4007/annals.2015.182.1.9},
      zblnumber = {06456013},
      }
  • [BD6] J. Bourgain and C. Demeter, Mean value estimates for Weyl sums in two dimensions, 2015.
    @misc{BD6,
      author = {Bourgain, Jean and Demeter, Ciprian},
      TITLE = {Mean value estimates for {W}eyl sums in two dimensions},
      arxiv = {1509.05388},
      year={2015},
      }
  • [BD4] J. Bourgain and C. Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, 2014.
    @misc{BD4,
      author = {Bourgain, Jean and Demeter, Ciprian},
      TITLE = {Decouplings for curves and hypersurfaces with nonzero {G}aussian curvature},
      NOTE={to appear in {\em J. d'Analyse Math.}},
      arxiv = {1409.1634},
      year={2014},
      }
  • [BD5] Go to document J. Bourgain and C. Demeter, "Decouplings for surfaces in $\mathbb{R}^4$," J. Funct. Anal., vol. 270, iss. 4, p. 1299–-1318, 2016.
    @article {BD5, MRKEY = {3447712},
      AUTHOR = {Bourgain, Jean and Demeter, Ciprian},
      TITLE = {Decouplings for surfaces in {$\mathbb{R}^4$}},
      JOURNAL = {J. Funct. Anal.},
      VOLUME = {270},
      YEAR = {2016},
      NUMBER = {4},
      PAGES = {1299–-1318},
      ISSN = {0022-1236},
      MRCLASS = {Preliminary Data},
      MRNUMBER = {3447712},
      DOI = {10.1016/j.jfa.2015.11.008},
      zblnumber = {06535745},
     }
  • [BW] J. Bourgain and N. Watt, Decoupling for perturbed cones and mean square of $\zeta(\frac12+it)$, 2015.
    @misc{BW,
      author = {Bourgain, Jean and Watt, N.},
      TITLE={Decoupling for perturbed cones and mean square of $\zeta(\frac12+it)$},
      arxiv = {1505.04161},
      year={2015},
      }
  • [BG] Go to document 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.
    @article {BG, MRKEY = {2860188},
      AUTHOR = {Bourgain, Jean and Guth, Larry},
      TITLE = {Bounds on oscillatory integral operators based on multilinear estimates},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {21},
      YEAR = {2011},
      NUMBER = {6},
      PAGES = {1239--1295},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {42B20},
      MRNUMBER = {2860188},
      MRREVIEWER = {Andrei K. Lerner},
      DOI = {10.1007/s00039-011-0140-9},
      zblnumber = {1237.42010},
      }
  • [FoWo] Go to document 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.
    @article {FoWo, MRKEY = {3286035},
      AUTHOR = {Ford, Kevin and Wooley, Trevor D.},
      TITLE = {On {V}inogradov's mean value theorem: strongly diagonal behaviour via efficient congruencing},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {213},
      YEAR = {2014},
      NUMBER = {2},
      PAGES = {199--236},
      ISSN = {0001-5962},
      MRCLASS = {11P05},
      MRNUMBER = {3286035},
      MRREVIEWER = {Karin Halupczok},
      DOI = {10.1007/s11511-014-0119-0},
      zblnumber = {1307.11102},
      }
  • [GarSe2] Go to document G. Garrigós and A. Seeger, "On plate decompositions of cone multipliers," Proc. Edinb. Math. Soc., vol. 52, iss. 3, pp. 631-651, 2009.
    @article {GarSe2, MRKEY = {2546636},
      AUTHOR = {Garrig{ó}s, Gustavo and Seeger, Andreas},
      TITLE = {On plate decompositions of cone multipliers},
      JOURNAL = {Proc. Edinb. Math. Soc.},
      FJOURNAL = {Proceedings of the Edinburgh Mathematical Society. Series II},
      VOLUME = {52},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {631--651},
      ISSN = {0013-0915},
      MRCLASS = {42B25},
      MRNUMBER = {2546636},
      MRREVIEWER = {Fr{é}d{é}ric Bernicot},
      DOI = {10.1017/S001309150700048X},
      zblnumber = {1196.42010},
      }
  • [Gu] Go to document L. Guth, "A short proof of the multilinear Kakeya inequality," Math. Proc. Cambridge Philos. Soc., vol. 158, iss. 1, pp. 147-153, 2015.
    @article {Gu, MRKEY = {3300318},
      AUTHOR = {Guth, Larry},
      TITLE = {A short proof of the multilinear {K}akeya inequality},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {158},
      YEAR = {2015},
      NUMBER = {1},
      PAGES = {147--153},
      ISSN = {0305-0041},
      MRCLASS = {42B08 (26D15)},
      MRNUMBER = {3300318},
      MRREVIEWER = {Timothy Michael Wertz},
      DOI = {10.1017/S0305004114000589},
      }
  • [PrSe] Go to document 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.
    @article {PrSe, MRKEY = {2288738},
      AUTHOR = {Pramanik, Malabika and Seeger, Andreas},
      TITLE = {{$L\sp p$} regularity of averages over curves and bounds for associated maximal operators},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {129},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {61--103},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {42B20},
      MRNUMBER = {2288738},
      MRREVIEWER = {Charles N. Moore},
      DOI = {10.1353/ajm.2007.0003},
      zblnumber = {1161.42009},
      }
  • [TWol] Go to document T. Wolff, "Local smoothing type estimates on $L^p$ for large $p$," Geom. Funct. Anal., vol. 10, iss. 5, pp. 1237-1288, 2000.
    @article {TWol, MRKEY = {1800068},
      AUTHOR = {Wolff, T.},
      TITLE = {Local smoothing type estimates on {$L\sp p$} for large {$p$}},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {10},
      YEAR = {2000},
      NUMBER = {5},
      PAGES = {1237--1288},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {42B25 (35L05 42B15)},
      MRNUMBER = {1800068},
      MRREVIEWER = {Anthony Carbery},
      DOI = {10.1007/PL00001652},
      zblnumber = {0972.42005},
      }
  • [Wo] Go to document T. D. Wooley, "The cubic case of the main conjecture in Vinogradov’s mean value theorem," Adv. Math., vol. 294, pp. 532-561, 2016.
    @article{Wo,
      author = {Wooley, Trevor D.},
      TITLE = {The cubic case of the main conjecture in {V}inogradov's mean value theorem},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {294},
      pages = {532--561},
      year = {2016},
      mrnumber = {3479572},
      zblnumber = {06567880},
      doi = {10.1016/j.aim.2016.02.033},
      }
  • [Wo1] T. D. Wooley, Approximating the main conjecture in Vinogradov’s mean value Theorem, 2014.
    @misc{Wo1,
      author = {Wooley, Trevor D.},
      TITLE = {Approximating the main conjecture in {V}inogradov's mean value Theorem},
      YEAR={2014},
      arxiv = {1401.2932},
      NOTE={52. pp.},
     }
  • [Wo2] Go to document T. D. Wooley, Translation invariance, exponential sums, and Waring’s problem.
    @misc{Wo2,
      author = {Wooley, Trevor D.},
      TITLE = {Translation invariance, exponential sums, and {W}aring's problem},
      SORTYEAR = {2014},
      url ={http://www.icm2014.org/download/Proceedings_Volume_II.pdf},
      note = {to appear in {\em Proc. Internat. Congress Math.},
      August 13--21 (Seoul, 2014), Volume II, Kyung Moon Sa Co. Ltd., Seoul, Korea pp. 505--529},
     }
  • [Wo3] Go to document T. D. Wooley, "The asymptotic formula in Waring’s problem," Int. Math. Res. Not., vol. 2012, iss. 7, pp. 1485-1504, 2012.
    @article {Wo3, MRKEY = {2913181},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {The asymptotic formula in {W}aring's problem},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2012},
      NUMBER = {7},
      PAGES = {1485--1504},
      ISSN = {1073-7928},
      MRCLASS = {11P05 (11P55)},
      MRNUMBER = {2913181},
      MRREVIEWER = {S. W. Graham},
      doi = {10.1093/imrn/rnr074},
      VOLUME = {2012},
      zblnumber = {1267.11104},
      }
  • [Wool1] Go to document T. D. Wooley, Discrete Fourier restriction via efficient congruencing.
    @misc{Wool1,
      author = {Wooley, Trevor D.},
      TITLE = {Discrete {F}ourier restriction via efficient congruencing},
      SORTYEAR={2016},
      note={{\em Internat. Math. Res. Notices},
      48 pp.},
      doi = {10.1093/imrn/rnw031},
     }

Keywords
, , ,
Mathematical Subject Classification
Primary: ,
DOI
https://doi.org/10.4007/annals.2016.184.2.7
MR
3548534
zbMATH
06662221
Milestones
Received: 14 December 2015
Revised: 18 April 2016
Accepted: 1 February 2016
Published online: 29 July 2016

Authors

Jean Bourgain

School of Mathematics, Institute for Advanced Study, Princeton, NJ

Ciprian Demeter

Department of Mathematics, Indiana University, Bloomington, IN

Larry Guth

Department of Mathematics, MIT, Cambridge, MA