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    $\begingroup$ thank for the explanation and the references. I appreciate your kind tone and I will try to adapt the proof to Waring's problem. $\endgroup$ Commented 10 hours ago
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    $\begingroup$ Dear GH: I wonder the degree to which "(which is still the only method known for the problem)" is true. It depends, of course, on how widely you define the circle method, but there are proofs of Vinogradov's theorem in the literature which do not use the circle method in its classical form -- I have in mind MR3347954 and MR0834356. The latter especially does not seem to use anything I would consider a circle method on a quick skim. $\endgroup$ Commented 10 hours ago
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    $\begingroup$ @AnuragSahay I did not know about these references, thanks for pointing them out! I updated my post accordingly. $\endgroup$ Commented 9 hours ago