Trace bounds for high powers in the singular value method

Derive good upper bounds for tr((Mw Mw)^r) when r ≥ 4, where Mw is the |W| × N matrix with entries Mw_{t,n} = w(n/N) e^{it log n} for t ∈ W and n ~ N, with W a Te-separated subset of an interval of length T and w a fixed smooth bump supported on [1, 2], to enable improved control of the largest singular value s1(Mw) beyond the current r = 3 approach.

Background

A central step in the authors’ method bounds the largest singular value s1(Mw) via traces of powers of Mw*Mw. Sharp control of tr((Mw*Mw)^r) for larger r would potentially yield stronger bounds on s1(Mw) and thus on the frequency of large values of Dirichlet polynomials, directly impacting zero-density estimates and prime distribution results.

The authors note that while r = 3 can be handled with their techniques, extending accurate trace bounds to r ≥ 4 remains unresolved, and such progress could bridge the gap to conjecturally optimal estimates aligned with Montgomery’s conjecture.