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Decoupling, exponential sums and the Riemann zeta function
theorem_6: Theorem 6 of arXiv 1408.5794v2 gives the exponent pair with coordinates 13/84 plus epsilon and 55/84 plus epsilon, under the usual smooth-phase convention.
Jean Bourgain, Decoupling, exponential sums and the Riemann zeta function, Journal of the American Mathematical Society 30 (2017), 205–224, DOI 10.1090/jams/860. The edition read is arXiv:1408.5794v2, revised 22 February 2016. The arXiv record specifies its non-exclusive distribution license, which does not grant open redistribution. No file of this source is held.
The paper uses decoupling to estimate a mean value arising in the Bombieri–Iwaniec method. The relevant reusable conclusion is Theorem 6, in Section 5: for each , is an exponent pair. Its use requires the usual phase-class hypotheses, including control of sufficiently many derivatives. Bounds on only a few derivatives do not by themselves provide that interface.
Read status: claims checked. The introduction and all of Section 5 were read in the arXiv v2 TeX source. Theorem 6 and its reduction to complete intervals and the complementary parameter range were read for their hypotheses and argument structure. The decoupling proof, the earlier exponential-sum estimates, and the cited exponent-pair theory were not independently verified. This is no proof or original-PDF fidelity warrant.
Bears on. Problem 962: Theorem 6 is a source for stronger exponential-sum bounds in a uniform smooth-integer interval argument. A deduction must check the complete phase class and retain the uniformity of its constants; the theorem alone does not give the conjectured subpower estimate.