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Sums of k-th powers in number fields

Part of:Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Morley Davidson
Morley Davidson
Affiliation:
Dept. of Math. and C.S., Kent State University, Kent, Ohio 44242, U.S.A.
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As usual in Waring's problem, we let G(k) be the least number s such that all sufficiently large natural numbers can be written as the sum of s or fewer k-th powers of positive integers. Hardy and Littlewood gave the first general upper bound, G(k)≤(k − 2)2k − 1 + 5. Later [4], they reduced this to order k2 by first constructing an auxiliary set of natural numbers below x which are sums of sk-th powers, of cardinality with for large k. The argument, which is brief and elementary (see [15, Chapter 6]), is now referred to by R. C. Vaughan's term “diminishing ranges”, since the k-th powers are taken from intervals of decreasing lengths. This idea of choosing the variables from restricted ranges was refined by Vinogradov, whose application to exponential sum estimates gave for large k (see [18]). The recent iterative method of Vaughan and Wooley [16, 19, 21], which halves Vinogradov's bound, may be viewed as an evolved diminishing ranges argument, producing an auxiliary set with .

MSC classification

Secondary:11P05: Waring's problem and variants

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Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998, pp. 359 - 370
Copyright
Copyright © University College London 1998

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