I'm currently conducting a bootstrap simulation for a difference of proportions hypothesis test (whether the proportion of Steph Curry's successful basketball shots from beyond the left side of the three-point line is equal to that of successful shots from beyond the right side).
In my lecture slides, the number of reps used is almost always equal to 1000, regardless of the size of the original sample, from which the resamples were drawn.
I am interested in knowing whether there is a way to mathematically justify the number of reps used in the simulation. I have looked online and found rules of thumb and the idea that once the distribution "stabilizes" the number of reps is enough, but it would be cool to know whether there is a way to concretely mathematically justify the number of reps or even determine an "optimal" number of reps that isn't infinity.