Acta Mathematica

On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing

Kevin Ford and Trevor D. Wooley

Full-text: Open access

Abstract

We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with 1sk2-1. In this way, we prove the main conjecture for such even moments when 1s14(k+1)2, showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of s, these finding application to the asymptotic formula in Waring’s problem.

Note

The first author was supported in part by National Science Foundation grants DMS-0901339 and DMS-1201442.

Note

The second author was supported in part by a Royal Society Wolfson Research Merit Award.

Article information

Source
Acta Math., Volume 213, Number 2 (2014), 199-236.

Dates
Received: 25 April 2013
Revised: 5 November 2013
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485801866

Digital Object Identifier
doi:10.1007/s11511-014-0119-0

Mathematical Reviews number (MathSciNet)
MR3286035

Zentralblatt MATH identifier
1307.11102

Rights
2014 © Institut Mittag-Leffler

Citation

Ford, Kevin; Wooley, Trevor D. On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing. Acta Math. 213 (2014), no. 2, 199--236. doi:10.1007/s11511-014-0119-0. https://projecteuclid.org/euclid.acta/1485801866


Export citation

References

  • Arkhipov, G. I. & Karatsuba, A.A., A new estimate of an integral of I. M. Vinogradov. Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), 751–762 (Russian); English translation in Math. USSR–Izv., 13 (1979), 52–62.
  • Ford K.B.: New estimates for mean values of Weyl sums. Int. Math. Res. Notices, 1995, 155–171 (1995)
  • Hua, L. K., Additive Theory of Prime Numbers. Translations of Mathematical Monographs, 13. Amer. Math. Soc., Providence, RI, 1965.
  • Linnik U.V.: On Weyl’s sums. Mat. Sbornik, 12(54), 28–39 (1943)
  • Rogers, L. J., An extension of a certain theorem in inequalities. Messenger of Math., 17 (1888), 145–150.
  • Tyrina, O. V., A new estimate for I. M. Vinogradov’s trigonometric integral. Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 363–378, 447 (Russian); English translation in Math. USSR–Izv., 30 (1988), 337—351.
  • Vaughan, R. C., The Hardy–Littlewood Method. Cambridge Tracts in Mathematics, 125. Cambridge Univ. Press, Cambridge, 1997.
  • Vaughan, R. C. & Wooley, T. D., A special case of Vinogradov’s mean value theorem. Acta Arith., 79 (1997), 193–204.
  • Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers. Tr. Mat. Inst. Steklova, 23 (1947), 1–109 (Russian); English translation as a book: Interscience, London–New York, 1954.
  • Wooley T.D.: Quasi-diagonal behaviour in certain mean value theorems of additive number theory. J. Amer. Math. Soc., 7 (1994), 221–245.
  • Wooley, T.D., A note on simultaneous congruences. J. Number Theory, 58 (1996), 288–297.
  • Wooley, T.D., Weyl’s inequality and exponential sums over binary forms. Funct. Approx. Comment. Math., 28 (2000), 83–95.
  • Wooley, T.D., The asymptotic formula in Waring’s problem. Int. Math. Res. Not. IMRN, 2012:7 (2012), 1485–1504.
  • Wooley, T.D., Vinogradov’s mean value theorem via efficient congruencing. Ann. of Math., 175 (2012), 1575–1627.
  • Wooley, T.D., Vinogradov’s mean value theorem via efficient congruencing, II. Duke Math. J., 162 (2013), 673–730.