Duke Mathematical Journal

On Vu's thin basis theorem in Waring's problem

Trevor D. Wooley

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V. Vu has recently shown that when k≥2 and s is sufficiently large in terms of k, then there exists a set X (k), whose number of elements up to t is smaller than a constant times (t log t)1/s, for which all large integers n are represented as the sum of s kth powers of elements of X (k) in order log n ways. We establish this conclusion with sk log k, improving on the constraint implicit in Vu's work which forces s to be as large as k48k. Indeed, the methods of this paper show, roughly speaking, that whenever existing methods permit one to show that all large integers are the sum of H(k) kth powers of natural numbers, then H(k)+2 variables suffice to obtain a corresponding conclusion for "thin sets," in the sense of Vu.

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Duke Math. J., Volume 120, Number 1 (2003), 1-34.

First available in Project Euclid: 16 April 2004

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Zentralblatt MATH identifier

Primary: 11P05: Waring's problem and variants O5D40


Wooley, Trevor D. On Vu's thin basis theorem in Waring's problem. Duke Math. J. 120 (2003), no. 1, 1--34. doi:10.1215/S0012-7094-03-12011-6. https://projecteuclid.org/euclid.dmj/1082138623

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