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Additive representation in short intervals, I: Waring's problem for cubes

Published online by Cambridge University Press:  04 December 2007

J. Brüdern  and
T. D. Wooley
J. Brüdern
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Postfach 80 11 40, D-70511 Stuttgart, [email protected]
T. D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, [email protected]
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Abstract

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Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

Keywords

Waring's problemexceptional setsshort intervals11P0511P55
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 5 , September 2004 , pp. 1197 - 1220
Copyright
Foundation Compositio Mathematica 2004