How can one generate an open-ended sequence of low-discrepancy points in 3D? I'd like a low-discrepancy sequence of points over a 3D-hypercube $[-1,1]^3$, but don't want to have to commit to a fixed number $n$ of points beforehand, that is  just see how the numerical integration estimates develop with increasing numbers of low-discrepancy points.
I'd like to avoid have to start all over again, if the results with a fixed $n$ are unsatisfactory. Of course, one could just employ random numbers, but then the convergence behavior would be poorer.
"A sequence of n-tuples that fills n-space more uniformly than uncorrelated random points, sometimes also called a low-discrepancy sequence. Although the ordinary uniform random numbers and quasirandom sequences both produce uniformly distributed sequences, there is a big difference between the two." (mathworld.wolfram.com/QuasirandomSequence.html)
This question has also just been put on the mathematica.stack.exchange
(https://mathematica.stackexchange.com/questions/143457/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d)
Since in his answer below, Martin Roberts advances a very interesting, appealing approach to the open-ended low-discrepancy problem, I’d like to indicate an (ongoing) implementation of his approach I’ve just reported in https://arxiv.org/abs/1809.09040 . In sec. XI (p. 19) and Figs. 5 and 6 there, I analyze two problems—one with sampling dimension $d=36$ and one with $d=64$—both using the parameter $\bf{\alpha}_0$ set to 0 and also to $\frac{1}{2}$. To convert the quasi-uniformly distributed points yielded by the Roberts’ algorithm to quasi-uniformly distributed normal variates, I use the code developed by Henrik Schumacher in his answer to https://mathematica.stackexchange.com/questions/181099/can-i-use-compile-to-speed-up-inversecdf

another good solution to get an open-ended sequence is using the Halton method. It is also very easy to implement, even for any dimension! For d<8 it has usually good properties, beyond this more difficult will typically outperform Halton.
