Sublogarithmic interval-discrepancy rates in dimension one

Determine whether the logarithmic interval-discrepancy bound for generating Delone realizations of essentially free ergodic flows can be improved to an intermediate sublogarithmic rate.

Background

In dimension one, the paper constructs generating Delone realizations whose discrepancy on every bounded interval is bounded by Cρ log(2+|I|). It also proves that a uniformly bounded discrepancy bound would force the intensity to be a nonzero Koopman eigenvalue, ruling out such a bound for weakly mixing flows.

The authors state that their argument does not establish optimality of the logarithmic rate. The unresolved problem is to determine whether some strictly sublogarithmic, but not necessarily uniformly bounded, discrepancy rate can be achieved in the same general realization framework.