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?



Efficient boxing for a mean value in the Bombieri Iwaniec method
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. 


