It seems like the generally accepted answer to, "How many bootstrap replications should I run?" has been, "It depends." It depends seems like a correct answer. Nevertheless, I was wondering whether there might be a parallel between bootstrap observations and samples taken from a continuous process in such a way as to allow for a better answer.
Specifically, the Nyquist theorem appears to suggest one sample at at least twice the occupied bandwidth. Density plots of observed (or bootstrapped) distributions also have 'bandwidth'. Can these two types of bandwidth be equated for the purposes of specifying a desired degree of accuracy? That is, if attempting to get a bootstrapped distribution that is sufficient (maybe excessive) could one terminate once one had no fewer than two observations within each (perhaps rolling?) bin that is the desired degree of accuracy?
For point estimates it would then seem to be fully sufficient (maybe excessive) to sample until one collects two bootstrap samples within a minimum bin size of interest, e.g. if one wanted a score that was accurate to the hundredths place, then the estimate and its two nearest-sampled neighbors should be at least within .005 of each other – of course that assumes that you know where your point estimate will be, which you never will. So then, if you had a prior point estimate, you could wait until that condition was satisfied and then sample twice again as much to be 'sure'. Alternatively, if you really care about the fidelity of the entire distribution, then you could sample until no two observations are further apart than your nearest-neighbor threshold (eliminating observations that are identical). Of course it would be important to consider that some thresholds can never be met (e.g. a bootstrap of the mean of 1, 2, and 3 below ⅓).
This (of course) doesn't cover all types of bootstraps that one might want to perform, but it seems like it might cover some. This might end up being a matter of opinion, but I'm hoping there is an answer or informed opinion sufficient to keep the question alive.