## Duke Mathematical Journal

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

Trevor D. Wooley

#### Abstract

V. Vu has recently shown that when k≥2 and s is sufficiently large in terms of k, then there exists a set $\mathfrak{X}$(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 $\mathfrak{X}$(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.

#### Article information

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

Dates
First available in Project Euclid: 16 April 2004

https://projecteuclid.org/euclid.dmj/1082138623

Digital Object Identifier
doi:10.1215/S0012-7094-03-12011-6

Mathematical Reviews number (MathSciNet)
MR2010732

Zentralblatt MATH identifier
1047.11094

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

#### Citation

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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