[alThor]

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The solution for M must have had a hidden assumption that half the integers are negative?

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This assumption is encoded in the fact that I used a kind of principal value limit. Let F(L) denote the fraction of sites between -L and L which the frog misses. Then F(L) -> 1 - M as L -> infinity. If you take, for example, the limit of the fraction of missed sites in the box [-L,2L], you will get a different limit. In this sense, the "fraction of all integers" is another ambiguity in the problem statement.

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OK, I see it. (I'm glad my question was understandable.)

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Since I seem to be the odd man out in my interpretation, perhaps you could help me understand yours. Specifically, where does the frog start in your interpretation? Does he start at the origin? Does he start randomly and, if so, with what distribution? Does he start "at -infinity" and, if so, what does that mean formally? Does he even start at time n = 0, or does time extend infinitely in both directions?

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Just to add my vote, I was thinking to start him at any finite point (arbitrarily 0) at time zero, measure time forward, ask "what percentage of points in [0,X] get hit", and let X go to infty.