Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-23T12:18:04.931Z Has data issue: false hasContentIssue false

DAVENPORT’S METHOD AND SLIM EXCEPTIONAL SETS: THE ASYMPTOTIC FORMULAE IN WARING’S PROBLEM

Published online by Cambridge University Press:  13 July 2010

Koichi Kawada
Affiliation:
Department of Mathematics, Faculty of Education, Iwate University, Morioka 020-8550, Japan (email: kawada@iwate-u.ac.jp)
Trevor D. Wooley
Affiliation:
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, U.K. (email: matdw@bristol.ac.uk)
Get access

Abstract

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.

Type
Research Article
Copyright
Copyright © University College London 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Boklan, K. D., The asymptotic formula in Waring’s problem. Mathematika 41 (1994), 329347.CrossRefGoogle Scholar
[2]Brüdern, J., Kawada, K. and Wooley, T. D., Additive representation in thin sequences, I: Waring’s problem for cubes. Ann. Sci. École Norm. Sup. (4) 34 (2001), 471501.CrossRefGoogle Scholar
[3]Brüdern, J. and Wooley, T. D., The Hasse principle for pairs of diagonal cubic forms. Ann. of Math. (2) 166 (2007), 865895.CrossRefGoogle Scholar
[4]Brüdern, J. and Wooley, T. D., The asymptotic formulae in Waring’s problem for cubes. J. Reine Angew. Math. (in press).Google Scholar
[5]Davenport, H., On sums of positive integral kth powers. Amer. J. Math. 64 (1942), 189198.CrossRefGoogle Scholar
[6]Ford, K. B., New estimates for mean values of Weyl sums. Int. Math. Res. Not. (1995), 155171.CrossRefGoogle Scholar
[7]Heath-Brown, D. R., Weyl’s inequality, Hua’s inequality and Waring’s problem. J. London Math. Soc. (2) 38 (1988), 216230.CrossRefGoogle Scholar
[8]Hooley, C., On the representations of a number as the sum of four cubes: I. Proc. Lond. Math. Soc. (3) 36 (1978), 117140.CrossRefGoogle Scholar
[9]Kawada, K. and Wooley, T. D., Slim exceptional sets for sums of fourth and fifth powers. Acta Arith. 103 (2002), 225248.CrossRefGoogle Scholar
[10]Kawada, K. and Wooley, T. D., Relations between exceptional sets for additive problems. J. London Math. Soc. (2) (to appear).Google Scholar
[11]Parsell, S. T., The density of rational lines on cubic hypersurfaces. Trans. Amer. Math. Soc. 352 (2000), 50455062.CrossRefGoogle Scholar
[12]Vaughan, R. C., On Waring’s problem for cubes. J. Reine Angew. Math. 365 (1986), 122170.Google Scholar
[13]Vaughan, R. C., On Waring’s problem for smaller exponents. II. Mathematika 33 (1986), 622.CrossRefGoogle Scholar
[14]Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn., Cambridge University Press (Cambridge, 1997).CrossRefGoogle Scholar
[15]Wooley, T. D., On Vinogradov’s mean value theorem. Mathematika 39 (1992), 379399.CrossRefGoogle Scholar
[16]Wooley, T. D., Sums of three cubes. Mathematika 47 (2000), 5361.CrossRefGoogle Scholar
[17]Wooley, T. D., Slim exceptional sets for sums of four squares. Proc. Lond. Math. Soc. (3) 85 (2002), 121.CrossRefGoogle Scholar
[18]Wooley, T. D., Slim exceptional sets for sums of cubes. Canad. J. Math. 54 (2002), 417448.CrossRefGoogle Scholar
[19]Wooley, T. D., Slim exceptional sets in Waring’s problem: one square and five cubes. Q. J. Math. 53 (2002), 111118.CrossRefGoogle Scholar
[20]Wooley, T. D., Slim exceptional sets and the asymptotic formula in Waring’s problem. Math. Proc. Cambridge Philos. Soc. 134 (2003), 193206.CrossRefGoogle Scholar