If the paces are linearly independent over the rational numbers (which is true for "almost all" n-tuples of real numbers), isn't this problem easy? In the theory of almost periodic functions, there's a theorem about the fact that you can treat periodic functions with linearly independent frequencies as statistically independent for the purposes of calculating long-time-average probabilities.

I must be missing something.

A similar theorem exists in probability theory, called the Lovasz local lemma.

I think the goal of rjpilton's analysis was to do things using elementary methods. That said, you might be able to use your approach for a quick and dirty approximation.

I thought this was about an Iron Maiden song.