Criterion for coverings at the exact order ρ_H

Develop a criterion to decide whether the covering property (*)—existence, for every sufficiently large N, of a covering {[c_j,d_j]}_{j=1}^{k(N)} with k(N) ≤ N and \sum_{j=1}^{k(N)} √{det Ω_H(c_j,d_j)} \lesssim N^{1−1/α}—is satisfied when α equals the order ρ_H of the monodromy matrix (not only for α > ρ_H).

Background

Romanov’s second theorem characterises the monodromy order ρ_H as the infimum of exponents α for which the interval admits coverings with controlled sums of √{det Ω_H}. Explicit coverings exist for all α > ρ_H, but checking the borderline case α = ρ_H is open.

A usable criterion at the exact order would refine the characterisation of ρ_H and may lead to sharper growth bounds.