It’s true that this technically answers the question in the negative, but one could still hope that if one imposes stronger conditions on the Beurling primes (for instance, some uniform spacing between Beurling integers) then one could use decoupling methods to improve upon the classical zero free region. I haven’t looked into this carefully, though. [Admittedly, the uniform spacing hypothesis is extremely strong and it may well be that it excludes all examples that aren’t essentially arithmetic in nature, but still it would emphasise the point that the Vinogradov zero free region is not inherently arithmetic in nature.]

]]>Diamond, Harold G.(1-IL); Montgomery, Hugh L.(1-MI); Vorhauer, Ulrike M. A.(1-KNTS) Beurling primes with large oscillation.

in which they show that 1 – c / \log t is the best zero-free region one can get in the case of Beurling primes (one can construct sets of Beurling primes with zeros beyond this region).

]]>*[Corrected, thanks – T.]*

Is it possible to combine both methods (to a possibly stronger method)?

]]>A key input of decoupling is the ability to induct on multiple scales, and we don’t seem to have an analogue of this in finite fields (where there are basically only two scales, the scale of a single point, and the scale of the entire field), except possibly in those situations where the finite field has a number of non-trivial subfields. On the other hand, Wooley has a recent preprint at https://arxiv.org/abs/1708.01220 where he develops a nested efficient congruencing method that is in some sense a “p-adic analogue” of decoupling (and in particular provides a second proof of the Vinogradov main conjecture) and which can be applied relatively easily to function field settings.

]]>I wonder if some of the decoupling techniques may be applicable in the finite field setting (it’s more of a blind guess rather than suggestion)?

]]>I’m currently studying the latest paper of Bourgain and Demeter on decoupling (https://arxiv.org/abs/1604.06032), but there is one issue I just don’t understand. In the proof of Prop. 8.4 in order to apply Thm. 5.1 the authors claim that the Fourier transform of the function (let’s call this function for fixed is supported in the neighborhood of the parabola . But for Thm. 5.1 I need that the Fourier support lies in the smaller set for some constant .

My question is the following: How is the Fourier transform of defined in order to give the author’s claim above any sense? Since does not lie in any space, the usual definition does not make sense, does it?

If I consider as a distribution, then I’m able to conclude that it has Fourier support (in the distributional sense) in . But that doesn’t seem to help me when I want to apply Thm. 5.1.

May I ask you to elaborate on this issue and explain how equation (21) in the above paper can be derived?

]]>