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One of the nice applications of decoupling is Bourgain’s record towards Lindelöf:

https://arxiv.org/pdf/1408.5794.pdf

Wooley has developed some techniques known as efficient congruencing which allow one to obtain estimates also derived from decoupling.

Lets call ’efficient boxing’ using Wooley’s efficient congruencing methods with archimedian metric instead of p-adic.

My question is: It is possible to use efficient boxing to recover the mean value estimate from the above Bourgain paper?

Some background about this: A direction along these lines was initiated by Watt

https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-39.3.385

so a similar question: Can Wooley’s innovations be combined with Watt’s ideas to recover Bourgain’s result?

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It seems possible since (p-adic) Efficient Congruencing also delivered the Optimal Estimate for Vinogradov's Mean Value Theorems. Another reason that this may be possible is that, if I recall correctly, historically inefficient boxing was introduced by Vinogradov and called the "short intervals method". This was applied to give bounds on the zeta function. A latter reference for this is Titchmarsh's Riemann Zeta-function book.

To get started I think one would combine a suitable adaptation of Wooley's "Nested Efficient Congruencing", R. Steiner's paper (https://arxiv.org/abs/1603.02536) for efficient boxing and Bourgain's paper.

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  • $\begingroup$ Many thanks for the reference. Do you also think there’s any way efficient congruencing with p-adic would work? $\endgroup$
    – zz7948
    May 15, 2020 at 5:16
  • $\begingroup$ Morally, if you can do something in one place, e.g. the infinite place which corresponds to the archimedean metric, then we should be able to do it in any other place, e.g. any p-adic metric. That said, it may be unclear how to adapt from one place to another. Looking at Bourgain's paper, at first glance I am uncertain how to interpret n^{3/2} p-adically. I think this issue is only psychological and that there is some p-adic version or approximation. Instead an essential point in Bourgain's paper is the non-vanishing Wronskian condition on his curves. $\endgroup$
    – K Hughes
    May 18, 2020 at 14:52
  • $\begingroup$ (above comment continued) Wooley has already adapted his efficient congruencing to curves with non-vanishing Wronskian. So I think that there is some hope to give a p-adic version. For instance, a thorough understanding - much better than I currently have - of Wooley's nested efficient congruencing paper would probably indicate how to interpret, p-adically, such expressions as (2.11) on page 11 of the Bourgain paper that you referenced. $\endgroup$
    – K Hughes
    May 18, 2020 at 14:52
  • $\begingroup$ Great, Thanks again. Efficient boxing does seem more natural for this problem although a p-adic approach would be nice $\endgroup$
    – zz7948
    May 20, 2020 at 10:36

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