On Waring's problem: Three cubes and a sixth power

J Brudern*, TD Wooley

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)

34 Citations (Scopus)

Abstract

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.
Translated title of the contributionOn Waring's problem: Three cubes and a sixth power
Original languageEnglish
Pages (from-to)13 - 53
Number of pages41
JournalNagoya Mathematical Journal
Volume163
DOIs
Publication statusPublished - Sep 2001

Bibliographical note

Publisher: Nagoya Univ, Dept Math - Faculty Sci

Keywords

  • IMPROVEMENTS
  • SUMS

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