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On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing

Kevin Ford Department of Mathematics, University of Illinois at Urbana-Champaign Trevor D. Wooley School of Mathematics, University of Bristol

Number Theory mathscidoc:1701.24005

Acta Mathematica, 213, (2), 199-236, 2013.4
We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2$s$, with $${1\leqslant s\leqslant k^{2}-1}$$ . In this way, we prove the main conjecture for such even moments when $${1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}$$ , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of$s$, these finding application to the asymptotic formula in Waring’s problem.
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@inproceedings{kevin2013on,
  title={On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing},
  author={Kevin Ford, and Trevor D. Wooley},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203403636815795},
  booktitle={Acta Mathematica},
  volume={213},
  number={2},
  pages={199-236},
  year={2013},
}
Kevin Ford, and Trevor D. Wooley. On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing. 2013. Vol. 213. In Acta Mathematica. pp.199-236. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203403636815795.
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