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It seems to be word in the generic corridor that decoupling (as in Bourgain-Demeter-Guth) and efficient congruencing (Wooley) are deeply related, and even that they are deep down the same thing - with efficient congruencing being the p-adic approach and decoupling the real one.

Has this remained at a pure "moral" level - an impressionistic statement based on many similarities? Or is it an insight that has given rise to an actual dictionary between the two approaches? What has been actually carried out?

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    $\begingroup$ One surprising thing about this is that we have the opportunity to prove "the same" theorem by both real and p-adic methods. Normally, proving a deep result over the integers requires local calculations at every place, but here we get away with proving the estimate with only calculations at one place, and not only that, but a place of our choice. $\endgroup$
    – Will Sawin
    Dec 18, 2017 at 19:14
  • $\begingroup$ Please use high level tags like "nt.number-theory" so that we don't miss your posts! $\endgroup$
    – GH from MO
    Jan 19, 2018 at 21:19

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I have a graduate student working on exactly this right now. At present, the student has a proof of two-dimensional decoupling theorems "in the style of Wooley's argument" (replacing p-adic concepts by their Archimedean counterparts, and largely following the presentation of Pierce mentioned in the other answer), and is currently looking at the higher dimensional case. The Wooley and Bourgain-Demeter-Guth arguments are not quite identical, but share many features in common, in particular a reliance on multi-scale iterative analysis and parabolic rescaling, and a fair amount of interpolation using Holder's inequality. (There are however some slight differences: Bourgain-Demeter-Guth's argument explicitly relies on a multilinear Kakeya inequality which appears to be absent (or very well hidden) in Wooley's argument, while Wooley's argument seems to give slightly better quantitative bounds but is (as of our current understanding of the argument) restricted to exponents that are even integers.) I expect his thesis will contain a large part of the dictionary you are looking for, but he has another year before he finishes.

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    $\begingroup$ Is the two dimensional case already on Arxiv? $\endgroup$
    – Fan Zheng
    Dec 19, 2017 at 0:22
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    $\begingroup$ No, but probably this will also be part of my student's thesis. $\endgroup$
    – Terry Tao
    Dec 19, 2017 at 2:45
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Some attempts at answering this can be found in the recent Bourbaki report of L. Pierce.

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  • $\begingroup$ I've been reading it together with my students and postdocs! We have seen some parallels, yes, but there is work to be done. $\endgroup$
    – H A Helfgott
    Dec 18, 2017 at 21:57
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Shaoming Guo, Zane Kun Li, Po-Lam Yung, and I found a bilinear proof of the decoupling inequality for the moment curve that uses the same inductive procedure as Wooley's efficient congruencing proof (Zane was the graduate student mentioned in Terry's answer from 2017).

The overall structure of the (Bourgain-Demeter-Guth) multilinear and the bilinear proofs of decoupling turned out to be quite similar, and both can be written concisely using some definitions introduced in a blog post by Tao. However, as opposed to the multilinear argument, we do not use any Kakeya-type inequalities.

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