Nagoya Mathematical Journal

On Waring’s problem: Three cubes and a minicube

Jörg Brüdern and Trevor D. Wooley

Full-text: Open access

Abstract

We establish that almost all natural numbers n are the sum of four cubes of positive integers, one of which is no larger than n5/36. The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

Article information

Source
Nagoya Math. J., Volume 200 (2010), 59-91.

Dates
First available in Project Euclid: 28 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1293500427

Digital Object Identifier
doi:10.1215/00277630-2010-012

Mathematical Reviews number (MathSciNet)
MR2747878

Zentralblatt MATH identifier
1238.11090

Subjects
Primary: 11P05: Waring's problem and variants 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

Citation

Brüdern, Jörg; Wooley, Trevor D. On Waring’s problem: Three cubes and a minicube. Nagoya Math. J. 200 (2010), 59--91. doi:10.1215/00277630-2010-012. https://projecteuclid.org/euclid.nmj/1293500427


Export citation

References

  • [1] K. D. Boklan, A reduction technique in Waring’s problem, I, Acta Arith. 65 (1993), 147–161.
  • [2] J. Brüdern, A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103 (1988), 27–33.
  • [3] J. Brüdern, Sums of four cubes, Monatsh. Math. 107 (1989), 179–188.
  • [4] J. Brüdern, On Waring’s problem for cubes, Math. Proc. Cambridge Philos. Soc. 109 (1991), 229–256.
  • [5] J. Brüdern and T. D. Wooley, On Waring’s problem: Three cubes and a sixth power, Nagoya Math. J. 163 (2001), 13–53.
  • [6] H. Davenport, On Waring’s problem for cubes, Acta Math. 71 (1939), 123–143.
  • [7] H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd ed., Cambridge University Press, Cambridge, 2005.
  • [8] R. A. Hunt, “On the convergence of Fourier series” in Proceedings of the Conference on Orthogonal Expansions and Their Continuous Analogues (Edwardsville, Ill., 1967), Southern Illinois University Press, Carbondale, Illinois, 1968, 235–255.
  • [9] K. Kawada, On the sum of four cubes, Mathematika 43 (1996), 323–348.
  • [10] R. C. Vaughan, Sums of three cubes, Bull. Lond. Math. Soc. 17 (1985), 17–20.
  • [11] R. C. Vaughan, On Waring’s problem for cubes, J. Reine Angew. Math. 365 (1986), 122–170.
  • [12] R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), 1–71.
  • [13] R. C. Vaughan, On Waring’s problem for cubes, II, J. Lond. Math. Soc. (2) 39 (1989), 205–218.
  • [14] R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge University Press, Cambridge, 1997.
  • [15] T. D. Wooley, On simultaneous additive equations, II, J. Reine Angew. Math. 419 (1991), 141–198.
  • [16] T. D. Wooley, Breaking classical convexity in Waring’s problem: Sums of cubes and quasi-diagonal behaviour, Invent. Math. 122 (1995), 421–451.
  • [17] T. D. Wooley, Sums of three cubes, Mathematika 47 (2000), 53–61.
  • [18] T. D. Wooley, A light-weight version of Waring’s problem, J. Austral. Math. Soc. Ser. A 76 (2004), 303–316.