Optimality of the logarithmic rate

Determine whether the logarithmic rate in the bound 1+c log(r)/r for the ratio of the largest to smallest lengths of r consecutive intervals is optimal, particularly for the van der Corput sequence.

Background

The paper proves that both the van der Corput sequence in base 2 and the golden-ratio Kronecker sequence exhibit strong regularity at very short scales, yielding a ratio bounded above by 1+c log(r)/r. Brethouwer’s numerical work led to the conjecture that this logarithmic dependence on r may be the best possible rate.

The paper establishes the logarithmic upper rate but does not prove a matching lower bound showing that no smaller-order dependence on r can hold. Thus, the optimality of the logarithmic rate remains unresolved.