Vinogradov’s mean value theorem via efficient congruencing

Abstract

We obtain estimates for Vinogradov’s integral that for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of $s$ $k$th powers of natural numbers whenever $s\ge 2k^2+2k-3$.

  • [Ark1984] Go to document G. I. Arkhipov, "The Hilbert-Kamke problem," Izv. Akad. Nauk SSSR Ser. Mat., vol. 48, iss. 1, pp. 3-52, 1984.
    @article {Ark1984, MRKEY = {0733357},
      AUTHOR = {Arkhipov, G. I.},
      TITLE = {The {H}ilbert-{K}amke problem},
      JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {48},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {3--52},
      ISSN = {0373-2436},
      MRCLASS = {11P55 (11L40)},
      MRNUMBER = {0733357},
      MRREVIEWER = {Andr{á}s S{á}rk{ö}zy},
      ZBLNUMBER = {0539.10016},
      DOI = {10.1070/IM1985v024n01ABEH001210},
     }
  • [ACK] G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, vol. 39.
    @book {ACK, MRKEY = {2113479},
      AUTHOR = {Arkhipov, G. I. and Chubarikov, V. N. and Karatsuba, A. A.},
      TITLE = {Trigonometric Sums in Number Theory and Analysis},
      SERIES = {de Gruyter Exp. Math.},
      VOLUME = {39},
      PUBLISHER = {Walter de Gruyter GmbH \& Co. KG, Berlin},
      YEAR = {2004},
      PAGES = {x+554},
      ISBN = {3-11-016266-0},
      MRCLASS = {11L07 (11L20 11P55)},
      MRNUMBER = {2113479},
      MRREVIEWER = {D. R. Heath-Brown},
      ZBLNUMBER = {1074.11043},
      }
  • [Bak1986] R. C. Baker, Diophantine Inequalities, New York: The Clarendon Press Oxford University Press, 1986, vol. 1.
    @book {Bak1986, MRKEY = {0865981},
      AUTHOR = {Baker, R. C.},
      TITLE = {Diophantine Inequalities},
      SERIES = {London Math. Soc. Monogr. (N.S.)},
      VOLUME = {1},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1986},
      PAGES = {xii+275},
      ISBN = {0-19-853545-7},
      MRCLASS = {11D75 (11E10 11J54)},
      MRNUMBER = {0865981},
      MRREVIEWER = {Ming-Chit Liu},
      ZBLNUMBER = {0592.10029},
      }
  • [Bir1961] Go to document B. J. Birch, "Waring’s problem in algebraic number fields," Proc. Cambridge Philos. Soc., vol. 57, pp. 449-459, 1961.
    @article {Bir1961, MRKEY = {0143754},
      AUTHOR = {Birch, B. J.},
      TITLE = {Waring's problem in algebraic number fields},
      JOURNAL = {Proc. Cambridge Philos. Soc.},
      VOLUME = {57},
      YEAR = {1961},
      PAGES = {449--459},
      MRCLASS = {10.46 (10.65)},
      MRNUMBER = {0143754},
      MRREVIEWER = {G. J. Rieger},
      ZBLNUMBER = {0111.25104},
      DOI = {10.1017/S0305004100035490},
     }
  • [Bok1994] Go to document K. D. Boklan, "The asymptotic formula in Waring’s problem," Mathematika, vol. 41, iss. 2, pp. 329-347, 1994.
    @article {Bok1994, MRKEY = {1316613},
      AUTHOR = {Boklan, Kent D.},
      TITLE = {The asymptotic formula in {W}aring's problem},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {41},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {329--347},
      ISSN = {0025-5793},
      CODEN = {MTKAAB},
      MRCLASS = {11P05},
      MRNUMBER = {1316613},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1112/S0025579300007439},
      ZBLNUMBER = {0815.11050},
      }
  • [BK2010] K. D. Boklan and T. D. Wooley, On Weyl sums for smaller exponents.
    @misc{BK2010,
      author={Boklan, Kent D. and Wooley, T. D.},
      TITLE={On {W}eyl sums for smaller exponents},
      NOTE={{\it Funct. Approx. Comment. Math.},
      in press},
      SORTYEAR={2010},
      }
  • [Bru1988] Go to document J. Brüdern, "A problem in additive number theory," Math. Proc. Cambridge Philos. Soc., vol. 103, iss. 1, pp. 27-33, 1988.
    @article {Bru1988, MRKEY = {0913447},
      AUTHOR = {Br{ü}dern, J{ö}rg},
      TITLE = {A problem in additive number theory},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {103},
      YEAR = {1988},
      NUMBER = {1},
      PAGES = {27--33},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {11P05},
      MRNUMBER = {0913447},
      MRREVIEWER = {Sh{ô} Iseki},
      DOI = {10.1017/S0305004100064586},
      ZBLNUMBER = {0655.10041},
     }
  • [CH2010] Go to document E. Croot and D. Hart, "$h$-fold sums from a set with few products," SIAM J. Discrete Math., vol. 24, iss. 2, pp. 505-519, 2010.
    @article {CH2010, MRKEY = {2646099},
      AUTHOR = {Croot, Ernie and Hart, Derrick},
      TITLE = {{$h$}-fold sums from a set with few products},
      JOURNAL = {SIAM J. Discrete Math.},
      FJOURNAL = {SIAM Journal on Discrete Mathematics},
      VOLUME = {24},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {505--519},
      ISSN = {0895-4801},
      MRCLASS = {11B30 (11B75 11N60)},
      MRNUMBER = {2646099},
      MRREVIEWER = {Serge{\u\i} V. Konyagin},
      DOI = {10.1137/090756041},
      ZBLNUMBER = {1221.11202},
      }
  • [For1995] Go to document K. B. Ford, "New estimates for mean values of Weyl sums," Internat. Math. Res. Notices, iss. 3, pp. 155-171, 1995.
    @article {For1995, MRKEY = {1321702},
      AUTHOR = {Ford, Kevin B.},
      TITLE = {New estimates for mean values of {W}eyl sums},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1995},
      NUMBER = {3},
      PAGES = {155--171},
      ISSN = {1073-7928},
      MRCLASS = {11P05},
      MRNUMBER = {1321702},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1155/S1073792895000122},
      ZBLNUMBER = {0821.11050},
      }
  • [For2002] Go to document K. B. Ford, "Vinogradov’s integral and bounds for the Riemann zeta function," Proc. London Math. Soc., vol. 85, iss. 3, pp. 565-633, 2002.
    @article {For2002, MRKEY = {1936814},
      AUTHOR = {Ford, Kevin B.},
      TITLE = {Vinogradov's integral and bounds for the {R}iemann zeta function},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {85},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {565--633},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {11L15 (11D72 11M06 11M26 11M35 11Y35)},
      MRNUMBER = {1936814},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1112/S0024611502013655},
      ZBLNUMBER = {1034.11044},
      }
  • [HL1922] Go to document G. H. Hardy and J. E. Littlewood, "Some problems of `Partitio Numerorum’: IV. The singular series in Waring’s Problem and the value of the number $G(k)$," Math. Z., vol. 12, iss. 1, pp. 161-188, 1922.
    @article {HL1922, MRKEY = {1544511},
      AUTHOR = {Hardy, G. H. and Littlewood, J. E.},
      TITLE = {Some problems of `{P}artitio {N}umerorum': {IV}. {T}he singular series in {W}aring's {P}roblem and the value of the number {$G(k)$}},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {12},
      YEAR = {1922},
      NUMBER = {1},
      PAGES = {161--188},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1544511},
      DOI = {10.1007/BF01482074},
      JFMNUMBER = {48.0146.01},
     }
  • [HB1988] Go to document D. R. Heath-Brown, "Weyl’s inequality, Hua’s inequality, and Waring’s problem," J. London Math. Soc., vol. 38, iss. 2, pp. 216-230, 1988.
    @article {HB1988, MRKEY = {0966294},
      AUTHOR = {Heath-Brown, D. R.},
      TITLE = {Weyl's inequality, {H}ua's inequality, and {W}aring's problem},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {38},
      YEAR = {1988},
      NUMBER = {2},
      PAGES = {216--230},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {11P05},
      MRNUMBER = {0966294},
      MRREVIEWER = {Sh{ô} Iseki},
      DOI = {10.1112/jlms/s2-38.2.216},
      ZBLNUMBER = {0619.10046},
      }
  • [Hil1909] Go to document D. Hilbert, "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n^{ter}$ Potenzen (Waringsches Problem)," Math. Ann., vol. 67, iss. 3, pp. 281-300, 1909.
    @article {Hil1909, MRKEY = {1511530},
      AUTHOR = {Hilbert, David},
      TITLE = {Beweis für die {D}arstellbarkeit der ganzen {Z}ahlen durch eine feste {A}nzahl {$n\sp {ter}$} {P}otenzen ({W}aringsches {P}roblem)},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {67},
      YEAR = {1909},
      NUMBER = {3},
      PAGES = {281--300},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1511530},
      DOI = {10.1007/BF01450405},
      JFMNUMBER = {40.0236.03},
      }
  • [Hua1938] Go to document L. Hua, "On Tarry’s problem," Quart. J. Math. Oxford, vol. 9, pp. 315-320, 1938.
    @article{Hua1938,
      author={Hua, Loo-Keng},
      TITLE = {On {T}arry's problem},
      JOURNAL={Quart. J. Math. Oxford},
      VOLUME={9},
      YEAR={1938},
      PAGES={315--320},
      ZBLNUMBER = {0020.00501},
      DOI = {10.1093/qmath/os-9.1.315},
      }
  • [Hua1949a] Go to document L. Hua, "Improvement of a result of Wright," J. London Math. Soc., vol. 24, pp. 157-159, 1949.
    @article {Hua1949a, MRKEY = {0030980},
      AUTHOR = {Hua, Loo-Keng},
      TITLE = {Improvement of a result of {W}right},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {24},
      YEAR = {1949},
      PAGES = {157--159},
      ISSN = {0024-6107},
      MRCLASS = {10.0X},
      MRNUMBER = {0030980},
      MRREVIEWER = {D. H. Lehmer},
      ZBLNUMBER = {0038.18104},
      DOI = {10.1112/jlms/s1-24.2.157},
     }
  • [Hua1949b] Go to document L. Hua, "An improvement of Vinogradov’s mean-value theorem and several applications," Quart. J. Math., Oxford Ser., vol. 20, pp. 48-61, 1949.
    @article {Hua1949b, MRKEY = {0029415},
      AUTHOR = {Hua, Loo-Keng},
      TITLE = {An improvement of {V}inogradov's mean-value theorem and several applications},
      JOURNAL = {Quart. J. Math., Oxford Ser.},
      FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      VOLUME = {20},
      YEAR = {1949},
      PAGES = {48--61},
      ISSN = {0033-5606},
      MRCLASS = {10.0X},
      MRNUMBER = {0029415},
      MRREVIEWER = {L. Schoenfeld},
      ZBLNUMBER = {0039.27403},
      ZBLNUMBER = {0039.27403},
      DOI = {10.1093/qmath/os-20.1.48},
     }
  • [Hua1965] L. Hua, Additive Theory of Prime Numbers, Providence, R.I.: Amer. Math. Soc., 1965, vol. 13.
    @book {Hua1965, MRKEY = {0194404},
      AUTHOR = {Hua, Loo-Keng},
      TITLE = {Additive Theory of Prime Numbers},
      SERIES = {Transl. Math. Monogr.},
      VOLUME={13},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1965},
      PAGES = {xiii+190},
      MRCLASS = {10.47},
      MRNUMBER = {0194404},
      ZBLNUMBER = {0192.39304},
      }
  • [Kar1973] Go to document A. A. Karatsuba, "The mean value of the modulus of a trigonometric sum," Izv. Akad. Nauk SSSR Ser. Mat., vol. 37, pp. 1203-1227, 1973.
    @article {Kar1973, MRKEY = {0337817},
      AUTHOR = {Karatsuba, A. A.},
      TITLE = {The mean value of the modulus of a trigonometric sum},
      JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {37},
      YEAR = {1973},
      PAGES = {1203--1227},
      ISSN = {0373-2436},
      MRCLASS = {10G05},
      MRNUMBER = {0337817},
      MRREVIEWER = {A. I. Vinogradov},
      ZBLNUMBER = {0294.10025},
      DOI = {10.1070/IM1973v007n06ABEH002080},
      }
  • [Lin1943] Go to document . V. Linnik, "On Weyl’s sums," Rec. Math. [Mat. Sbornik] N.S., vol. 12(54), pp. 28-39, 1943.
    @article {Lin1943, MRKEY = {0009776},
      AUTHOR = {Linnik, {Yu} V.},
      TITLE = {On {W}eyl's sums},
      JOURNAL = {Rec. Math. [Mat. Sbornik] N.S.},
      VOLUME = {12(54)},
      YEAR = {1943},
      PAGES = {28--39},
      MRCLASS = {10.0X},
      MRNUMBER = {0009776},
      MRREVIEWER = {G. Pall},
      ZBLNUMBER = {0063.03578},
      URL = {http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=6141&option_lang=eng},
      }
  • [Mit1986] Go to document D. A. Mit$’$kin, "An estimate for the number of summands in the Hilbert-Kamke problem," Mat. Sb., vol. 129(171), iss. 4, pp. 549-577, 592, 1986.
    @article {Mit1986, MRKEY = {0842400},
      AUTHOR = {Mit$'$kin, D. A.},
      TITLE = {An estimate for the number of summands in the {H}ilbert-{K}amke problem},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {MatematicheskiĭSbornik. Novaya Seriya},
      VOLUME = {129(171)},
      YEAR = {1986},
      NUMBER = {4},
      PAGES = {549--577, 592},
      ISSN = {0368-8666},
      MRCLASS = {11D41 (11P55)},
      MRNUMBER = {0842400},
      MRREVIEWER = {J. Browkin},
      ZBLNUMBER = {0608.10020},
      DOI = {10.1070/SM1987v057n02ABEH003087},
      }
  • [Mit1987] Go to document D. A. Mit$’$kin, "An estimate for the number of summands in the Hilbert-Kamke problem. II," Mat. Sb., vol. 132(174), iss. 3, pp. 345-351, 444, 1987.
    @article {Mit1987, MRKEY = {0889596},
      AUTHOR = {Mit$'$kin, D. A.},
      TITLE = {An estimate for the number of summands in the {H}ilbert-{K}amke problem. {II}},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {MatematicheskiĭSbornik. Novaya Seriya},
      VOLUME = {132(174)},
      YEAR = {1987},
      NUMBER = {3},
      PAGES = {345--351, 444--445},
      ISSN = {0368-8666},
      MRCLASS = {11D41},
      MRNUMBER = {0889596},
      MRREVIEWER = {J. Browkin},
      ZBLNUMBER = {0619.10013},
      DOI = {10.1070/SM1988v060n02ABEH003172},
     }
  • [Par2005] Go to document S. T. Parsell, "A generalization of Vinogradov’s mean value theorem," Proc. London Math. Soc., vol. 91, iss. 1, pp. 1-32, 2005.
    @article {Par2005, MRKEY = {2149529},
      AUTHOR = {Parsell, Scott T.},
      TITLE = {A generalization of {V}inogradov's mean value theorem},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {91},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {1--32},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {11D45 (11D72 11L07 11P55)},
      MRNUMBER = {2149529},
      MRREVIEWER = {Rainer Dietmann},
      DOI = {10.1112/S002461150501525X},
      ZBLNUMBER = {1119.11024},
      }
  • [Par2009] Go to document S. T. Parsell, "On the Bombieri–Korobov estimate for Weyl sums," Acta Arith., vol. 138, iss. 4, pp. 363-372, 2009.
    @article {Par2009, MRKEY = {2534142},
      AUTHOR = {Parsell, Scott T.},
      TITLE = {On the {B}ombieri--{K}orobov estimate for {W}eyl sums},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {138},
      YEAR = {2009},
      NUMBER = {4},
      PAGES = {363--372},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11L15 (11L07 11P05 11P55)},
      MRNUMBER = {2534142},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.4064/aa138-4-7},
      ZBLNUMBER = {05615154},
      }
  • [RS2000] O. Robert and P. Sargos, "Un théorème de moyenne pour les sommes d’exponentielles. Application à l’inégalité de Weyl," Publ. Inst. Math. $($Beograd$)$, vol. 67(81), pp. 14-30, 2000.
    @article {RS2000, MRKEY = {1761299},
      AUTHOR = {Robert, O. and Sargos, P.},
      TITLE = {Un théorème de moyenne pour les sommes d'exponentielles. {A}pplication à l'inégalité de {W}eyl},
      JOURNAL = {Publ. Inst. Math. $($Beograd$)$},
      FJOURNAL = {Institut Mathématique. Publications. Nouvelle Série},
      VOLUME = {67(81)},
      YEAR = {2000},
      PAGES = {14-30},
      ISSN = {0350-1302},
      MRCLASS = {11L07 (11L15)},
      MRNUMBER = {1761299},
      MRREVIEWER = {Angel V. Kumchev},
      ZBLNUMBER = {1006.11046},
      }
  • [Sch1985] Go to document W. M. Schmidt, "The density of integer points on homogeneous varieties," Acta Math., vol. 154, iss. 3-4, pp. 243-296, 1985.
    @article {Sch1985, MRKEY = {0781588},
      AUTHOR = {Schmidt, Wolfgang M.},
      TITLE = {The density of integer points on homogeneous varieties},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {154},
      YEAR = {1985},
      NUMBER = {3-4},
      PAGES = {243--296},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {11D72 (11P55 11R09 11R45)},
      MRNUMBER = {0781588},
      MRREVIEWER = {D. J. Lewis},
      DOI = {10.1007/BF02392473},
      ZBLNUMBER = {0561.10010},
      }
  • [Ste1975] Go to document S. B. Stechkin, "Mean values of the modulus of a trigonometric sum," Trudy Mat. Inst. Steklov., vol. 134, pp. 283-309, 411, 1975.
    @article {Ste1975, MRKEY = {0396431},
      AUTHOR = {Stechkin, S. B.},
      TITLE = {Mean values of the modulus of a trigonometric sum},
      JOURNAL = {Trudy Mat. Inst. Steklov.},
      FJOURNAL = {Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova},
      VOLUME = {134},
      YEAR = {1975},
      PAGES = {283--309, 411},
      ISSN = {0371-9685},
      MRCLASS = {10G05},
      MRNUMBER = {0396431},
      MRREVIEWER = {Karl-Heinz Indlekofer},
      ZBLNUMBER = {0319.10045},
      URL = {http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=2719&option_lang=eng},
      }
  • [Ust1998] Go to document A. V. Ustinov, "On the number of summands in the asymptotic formula for the number of solutions of the Waring equation," Mat. Zametki, vol. 64, iss. 2, pp. 285-296, 1998.
    @article {Ust1998, MRKEY = {1681004},
      AUTHOR = {Ustinov, A. V.},
      TITLE = {On the number of summands in the asymptotic formula for the number of solutions of the {W}aring equation},
      JOURNAL = {Mat. Zametki},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki},
      VOLUME = {64},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {285--296},
      ISSN = {0025-567X},
      MRCLASS = {11P05 (11L07)},
      MRNUMBER = {1681004},
      MRREVIEWER = {S. Turj{á}nyi},
      DOI = {10.1007/BF02310311},
      ZBLNUMBER = {0923.11138},
      }
  • [Vau1986a] Go to document R. C. Vaughan, "On Waring’s problem for cubes," J. Reine Angew. Math., vol. 365, pp. 122-170, 1986.
    @article {Vau1986a, MRKEY = {0826156},
      AUTHOR = {Vaughan, R. C.},
      TITLE = {On {W}aring's problem for cubes},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {365},
      YEAR = {1986},
      PAGES = {122--170},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11P05 (11P55)},
      MRNUMBER = {0826156},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1515/crll.1986.365.122},
      ZBLNUMBER = {0574.10046},
      }
  • [Vau1986b] Go to document R. C. Vaughan, "On Waring’s problem for smaller exponents. II," Mathematika, vol. 33, iss. 1, pp. 6-22, 1986.
    @article {Vau1986b, MRKEY = {0859494},
      AUTHOR = {Vaughan, R. C.},
      TITLE = {On {W}aring's problem for smaller exponents. {II}},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {33},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {6--22},
      ISSN = {0025-5793},
      CODEN = {MTKAAB},
      MRCLASS = {11P05},
      MRNUMBER = {0859494},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1112/S0025579300013838},
      ZBLNUMBER = {0601.10037},
      }
  • [Vau1997] R. C. Vaughan, The Hardy-Littlewood Method, Second ed., Cambridge: Cambridge Univ. Press, 1997, vol. 125.
    @book {Vau1997, MRKEY = {1435742},
      AUTHOR = {Vaughan, R. C.},
      TITLE = {The {H}ardy-{L}ittlewood Method},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {125},
      EDITION = {Second},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      PAGES = {xiv+232},
      ISBN = {0-521-57347-5},
      MRCLASS = {11P55 (11L15 11P05)},
      MRNUMBER = {1435742},
      MRREVIEWER = {D. R. Heath-Brown},
      ZBLNUMBER = {0868.11046},
      }
  • [VW1997] R. C. Vaughan and T. D. Wooley, "A special case of Vinogradov’s mean value theorem," Acta Arith., vol. 79, iss. 3, pp. 193-204, 1997.
    @article {VW1997, MRKEY = {1438823},
      AUTHOR = {Vaughan, R. C. and Wooley, T. D.},
      TITLE = {A special case of {V}inogradov's mean value theorem},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {79},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {193--204},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11D72 (11L15 11P05)},
      MRNUMBER = {1438823},
      MRREVIEWER = {D. R. Heath-Brown},
      ZBLNUMBER = {0887.11042},
      }
  • [Vin1935] I. M. Vinogradov, "New estimates for Weyl sums," Dokl. Akad. Nauk SSSR, vol. 8, pp. 195-198, 1935.
    @article{Vin1935,
      author = {Vinogradov, I. M.},
      TITLE={New estimates for {W}eyl sums},
      JOURNAL={Dokl. Akad. Nauk SSSR},
      VOLUME={8},
      YEAR={1935},
      PAGES={195--198},
     }
  • [Vin1947] I. M. Vinogradov, "The method of trigonometrical sums in the theory of numbers," Trav. Inst. Math. Stekloff, vol. 23, p. 109, 1947.
    @article {Vin1947, MRKEY = {0029417},
      AUTHOR = {Vinogradov, I. M.},
      TITLE = {The method of trigonometrical sums in the theory of numbers},
      JOURNAL = {Trav. Inst. Math. Stekloff},
      VOLUME = {23},
      YEAR = {1947},
      PAGES = {109 pp},
      MRCLASS = {10.0X},
      MRNUMBER = {0029417},
      MRREVIEWER = {L. Schoenfeld},
      ZBLNUMBER = {0041.37002},
     }
  • [Vin1958] Go to document I. M. Vinogradov, "A new estimate of the function $\zeta (1+it)$," Izv. Akad. Nauk SSSR. Ser. Mat., vol. 22, pp. 161-164, 1958.
    @article {Vin1958, MRKEY = {0103861},
      AUTHOR = {Vinogradov, I. M.},
      TITLE = {A new estimate of the function {$\zeta (1+it)$}},
      JOURNAL = {Izv. Akad. Nauk SSSR. Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {22},
      YEAR = {1958},
      PAGES = {161--164},
      ISSN = {0373-2436},
      MRCLASS = {10.00},
      MRNUMBER = {0103861},
      MRREVIEWER = {H. Davenport},
      ZBLNUMBER = {0097.26302},
      URL = {http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3962&option_lang=eng},
      }
  • [Wal1963] A. Z. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin: , 1963, vol. 15.
    @book{Wal1963,
      author={Walfisz, A. Z.},
      TITLE={Weylsche Exponentialsummen in der neueren Zahlentheorie},
      SERIES={Math. Forsch.},
      VOLUME={15},
      ADDRESS={Berlin},
      YEAR={1963},
      ZBLNUMBER = {0146.06003},
      }
  • [Woo1992a] Go to document T. D. Wooley, "Large improvements in Waring’s problem," Ann. of Math., vol. 135, iss. 1, pp. 131-164, 1992.
    @article {Woo1992a, MRKEY = {1147960},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {Large improvements in {W}aring's problem},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {135},
      YEAR = {1992},
      NUMBER = {1},
      PAGES = {131--164},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11P05},
      MRNUMBER = {1147960},
      MRREVIEWER = {J{ö}rg Br{ü}dern},
      DOI = {10.2307/2946566},
      ZBLNUMBER = {0754.11026},
      }
  • [Woo1992] Go to document T. D. Wooley, "On Vinogradov’s mean value theorem," Mathematika, vol. 39, iss. 2, pp. 379-399, 1992.
    @article {Woo1992, MRKEY = {1203293},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {On {V}inogradov's mean value theorem},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {39},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {379--399},
      ISSN = {0025-5793},
      CODEN = {MTKAAB},
      MRCLASS = {11P05},
      MRNUMBER = {1203293},
      MRREVIEWER = {J{ö}rg Br{ü}dern},
      DOI = {10.1112/S0025579300015102},
      ZBLNUMBER = {0769.11036},
      }
  • [Woo1993] Go to document T. D. Wooley, "On Vinogradov’s mean value theorem, II," Michigan Math. J., vol. 40, iss. 1, pp. 175-180, 1993.
    @article {Woo1993, MRKEY = {1214062},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {On {V}inogradov's mean value theorem, {II}},
      JOURNAL = {Michigan Math. J.},
      FJOURNAL = {The Michigan Mathematical Journal},
      VOLUME = {40},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {175--180},
      ISSN = {0026-2285},
      MRCLASS = {11P05 (11M06)},
      MRNUMBER = {1214062},
      MRREVIEWER = {J{ö}rg Br{ü}dern},
      DOI = {10.1307/mmj/1029004681},
      ZBLNUMBER = {0805.11072},
      }
  • [Woo1994] Go to document T. D. Wooley, "Quasi-diagonal behaviour in certain mean value theorems of additive number theory," J. Amer. Math. Soc., vol. 7, iss. 1, pp. 221-245, 1994.
    @article {Woo1994, MRKEY = {1224595},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {Quasi-diagonal behaviour in certain mean value theorems of additive number theory},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {7},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {221--245},
      ISSN = {0894-0347},
      MRCLASS = {11L15 (11P05)},
      MRNUMBER = {1224595},
      MRREVIEWER = {J{ö}rg Br{ü}dern},
      DOI = {10.2307/2152728},
      ZBLNUMBER = {0786.11053},
      }
  • [Woo1995] Go to document T. D. Wooley, "New estimates for Weyl sums," Quart. J. Math. Oxford Ser., vol. 46, iss. 181, pp. 119-127, 1995.
    @article {Woo1995, MRKEY = {1326136},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {New estimates for {W}eyl sums},
      JOURNAL = {Quart. J. Math. Oxford Ser.},
      FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      VOLUME = {46},
      YEAR = {1995},
      NUMBER = {181},
      PAGES = {119--127},
      ISSN = {0033-5606},
      CODEN = {QJMAAT},
      MRCLASS = {11P05 (11L15)},
      MRNUMBER = {1326136},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1093/qmath/46.1.119},
      ZBLNUMBER = {0855.11043},
      }
  • [Woo1995a] Go to document T. D. Wooley, "Breaking classical convexity in Waring’s problem: sums of cubes and quasi-diagonal behaviour," Invent. Math., vol. 122, iss. 3, pp. 421-451, 1995.
    @article {Woo1995a, MRKEY = {1359599},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {Breaking classical convexity in {W}aring's problem: sums of cubes and quasi-diagonal behaviour},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {122},
      YEAR = {1995},
      NUMBER = {3},
      PAGES = {421--451},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11P05 (11P55)},
      MRNUMBER = {1359599},
      MRREVIEWER = {J{ö}rg Br{ü}dern},
      DOI = {10.1007/BF01231451},
      ZBLNUMBER = {0851.11055},
      }
  • [Woo1996] Go to document T. D. Wooley, "Some remarks on Vinogradov’s mean value theorem and Tarry’s problem," Monatsh. Math., vol. 122, iss. 3, pp. 265-273, 1996.
    @article {Woo1996, MRKEY = {1414861},
      AUTHOR = {Wooley, Trevor D.},
      TITLE = {Some remarks on {V}inogradov's mean value theorem and {T}arry's problem},
      JOURNAL = {Monatsh. Math.},
      FJOURNAL = {Monatshefte für Mathematik},
      VOLUME = {122},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {265--273},
      ISSN = {0026-9255},
      CODEN = {MNMTA2},
      MRCLASS = {11D72 (11L07 11P55)},
      MRNUMBER = {1414861},
      MRREVIEWER = {R. C. Baker},
      DOI = {10.1007/BF01320189},
      ZBLNUMBER = {0881.11043},
      }
  • [Woo2011] Go to document T. D. Wooley, "The asymptotic formula in Waring’s problem," Internat. Math. Res. Notices, vol. (2012), no. 7, pp. 1485-1504.
    @article{Woo2011,
      author={Wooley, Trevor D.},
      TITLE = {The asymptotic formula in {W}aring's problem},
      JOURNAL={Internat. Math. Res. Notices},
      VOLUME={(2012), {\rm no.~7}},
      PAGES={1485--1504},
      DOI = {10.1093/imrn/rnr074},
      SORTYEAR={2012},
     }
  • [Wri1948] Go to document E. M. Wright, "The Prouhet-Lehmer problem," J. London Math. Soc., vol. 23, pp. 279-285, 1948.
    @article {Wri1948, MRKEY = {0028858},
      AUTHOR = {Wright, E. M.},
      TITLE = {The {P}rouhet-{L}ehmer problem},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {23},
      YEAR = {1948},
      PAGES = {279--285},
      ISSN = {0024-6107},
      MRCLASS = {10.0X},
      MRNUMBER = {0028858},
      MRREVIEWER = {D. H. Lehmer},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0033.35102},
      DOI = {10.1112/jlms/s1-23.4.279},
     }

Authors

Trevor D. Wooley

School of Mathematics
University of Bristol
University Walk
Bristol BS8 1TW
United Kingdom