Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-21T11:03:31.209Z Has data issue: false hasContentIssue false
  • English
  • Français

A light-weight version of Waring's problem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  09 April 2009

Trevor D. Wooley
Trevor D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Ave, Ann Arbor, MI 48109-1109, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An asymptotic formula is established for the number of representations of a large integer as the sum of kth powers of natural numbers, in which each representation is counted with a homogeneous weight that de-emphasises the large solutions. Such an asymptotic formula necessarily fails when this weight is excessively light.

Keywords

Waring's problemapplications of the Hardy-Littlewood method

MSC classification

Secondary: 11P05: Waring's problem and variants 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 303 - 316
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Ford, K. B., ‘New estimates for mean values of Weyl sums’, Internat. Math. Res. Notices (1995), 155–171.CrossRefGoogle Scholar
[2]Heath-Brown, D. R., ‘Weyl's inequality, Hau's inequality, and Waring's problem’, J. London Math. Soc. (2) 38 (1988), 216–230.CrossRefGoogle Scholar
[3]Hau, L.-K., ‘On Waring's problem’, Quart. J. Math. Oxford 9 (1938), 199–202.Google Scholar
[4]Vaughan, R. C., The Hardy-Littlewood method, 2nd edition (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
[5]Vu, Van H., ‘On a refinement of Waring's problem’, Duke Math. J. 105 (2000), 107–134.CrossRefGoogle Scholar
[6]Wooley, T. D., ‘On Vinogradov's mean value theorem’, Mathematika 39 (1992), 379–399.CrossRefGoogle Scholar
[7]Wooley, T. D., ‘On Vu's thin basis theorem in Waring's problem’, Duke Math. J. 120 (2003), 1–34.CrossRefGoogle Scholar