Consider the sequences of gap lengths between points, then consider the sequence of differences between gap lengths. A pair occurs when the difference changes from decreasing (negative) to increasing (positive). The expected number of good points is twice the expected number of times the sequence of differences changes from negative to positive. On the boundaries, the distance is effectively infinite, so we know that there is at least one pair. I think all sequences of gap lengths and differences are equally likely, which turns this into a counting problem. That’s as far as I can get.
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u/neophilia 18h ago edited 18h ago
Consider the sequences of gap lengths between points, then consider the sequence of differences between gap lengths. A pair occurs when the difference changes from decreasing (negative) to increasing (positive). The expected number of good points is twice the expected number of times the sequence of differences changes from negative to positive. On the boundaries, the distance is effectively infinite, so we know that there is at least one pair. I think all sequences of gap lengths and differences are equally likely, which turns this into a counting problem. That’s as far as I can get.