An open question about powers of log(n) in quasi-Monte Carlo.

The commonly quoted error rate for QMC integration with an infinite low discrepancy sequence is $O(n^{-1}\log(n)^d)$. This holds uniformly over $d$ dimensional integrands of Hardy-Krause variation one when using $n$ evaluation points. Implicit in the rate is that for any sequence of QMC points the integrand can be chosen to depend on $n$. If we have a fixed integrand, should we expect to need those powers of $\log(n)$ as $n\to\infty$? That is, do those powers provide a realistic idea of the progress that the algorithm makes as $n\to\infty$? This paper poses an open question about whether there is any $f$ and any $(t,d)$-sequence for which the absolute error exceeds a positive multiple of $\log(n)^r/n$ infinitely often for some $r>1$. For $r=1$, such functions are known from the discrepancy literature even for $d=1$. An empirical search when $d=2$ for integrands designed to exploit known weaknesses in certain point sets showed no evidence that $r>1$ is needed. An example with $d=3$ and $n$ up to $2^{100}$ appears to require $r>1$. Of course neither of these resolve the open problem.