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.

References

If one could establish such a sharp bound on tr((Mw Mw)") for large r, this would give Conjecture 1.5. Unfortunately we do not know how to obtain good bounds when r ≥ 4, so we work with r = 3.

— New large value estimates for Dirichlet polynomials  (2405.20552 - Guth et al., 2024) in Section 4 (The matrix Mw and its singular values), discussion preceding Lemma 4.2

In a more classical language, these computations correspond to the fact that with current understanding of the large values of Dirichlet polynomials, we cannot rule out the scenario that, for bounded coefficients a_j(n),

\left|\sum_{n \sim X{\beta_j}} \frac{a_j(n)}{n{1/2+it}}\right| \geq X{\beta_j(1/2-3/10)}

for \gg X{3/10} one-separated points t \in [-X{\tau}, X{\tau}].

In order to obtain the exponent 3/5-\varepsilon in Theorem \ref{thm:intro-uniformity}, we would need to essentially have, for T = X{O(1)}, the bound

\int_{X\varepsilon}{X{\tau+\varepsilon}}\left|\sum_{n \sim X{1/4}} \frac{1}{n{1/2+it+iT}}\right|4 dt \ll \frac{X{1/2}}{\logA X},

and it is the above-mentioned scenario which prevents us from obtaining this—the coefficients a_j(n)=n{-iT} prevent us from using type I information once T is a largish power of X.

— Higher order uniformity of the primes and cancellation of the Möbius function in shorter intervals  (2610.09567 - Matomäki et al., 7 Oct 2026) in Remark 5.3, Section 5 (A Baker–Harman–Pintz type trilinear estimate)