Sums of three cubes, II

@article{Wooley2015SumsOT,
  title={Sums of three cubes, II},
  author={Trevor D. Wooley},
  journal={arXiv: Number Theory},
  year={2015}
}
  • T. Wooley
  • Published 6 February 2015
  • Mathematics
  • arXiv: Number Theory
Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding $X$ that are represented as the sum of three cubes of natural numbers. 
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References

SHOWING 1-10 OF 24 REFERENCES
ON WARING'S PROBLEM FOR CUBES AND SMOOTH WEYL SUMS
Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection,
Relations between exceptional sets for additive problems
We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with
The Hasse principle for pairs of diagonal cubic forms
By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic
The circle method and diagonal cubic forms
  • D. R. Heath-Brown
  • Mathematics, Biology
    Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 1998
TLDR
The Hardy–Littlewood circle method is used to investigate the number of integer zeros of diagonal cubic forms, and it is shown that there are O(P3+ϵ) zeros up to height P, for any ϵ>0.
Quasi-Diagonal Behaviour and Smooth Weyl Sums
Abstract. Estimates are provided for small moments of exponential sums over smooth numbers substantially sharper than available hitherto. These bounds arise from the author’s recent breaking of
A problem in additive number theory
The determination of the minimal s such that all large natural numbers n admit a representation as is an interesting problem in the additive theory of numbers and has a considerable literature, For
Correlation estimates for sums of three cubes
We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of
Waring’s problem for cubes
In his book Meditationes Algebraicae, published in 1770, Edward Waring stated without proof that every nonnegative integer is the sum of four squares, nine cubes, 19 fourth powers, and so on.
ON A DIAGONAL QUADRIC IN DENSE VARIABLES
  • E. Keil
  • Mathematics
    Glasgow Mathematical Journal
  • 2014
Abstract We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets $\mathcal{A}$ ⊂ ℤ and show quantitative bounds on the size of
On Waring's problem: Three cubes and a sixth power
The natural interpretation of even moments of exponential sums, in terms of the number of solutions of certain underlying diophantine equations, permits a rich interplay to be developed between
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2
3
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