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Suppose we have the integers from $1$ to $N$ lined out and a proportion $0 < c < 1$ of them that we are going to pick uniformly randomly without replacement (so say we pick $\lfloor cN \rfloor$ of them). For fixed $c,$ how does the size of the maximal gap between the picked integers grow as $N$ gets large (defined as the maximum difference between two consecutive integers when the picked integers are arranged from smallest to largest)? Do we have good asymptotic results on how the size of this maximal gap grows?

Hadi Khan
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  • Related: https://math.stackexchange.com/questions/2274049/a-broken-stick-with-n-breaks/2275147 – leonbloy Mar 14 '22 at 02:39
  • You might also want to include the gap from the smallest to $0$ and the largest to $N+1$ when finding the biggest of the gaps. Are you interested in the expected largest gap? – Henry Mar 14 '22 at 02:41

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A useful approximation for problems like this is to imagine that each number is picked with probability $c$ independently. This ignores the correlations that come because you have you defined the total number picked, but they becomes less important as $N$ gets large. The chance that a given number is the first of a run of at least $m$ is then $c^m$. The expected number of runs of at least $m$ (counting runs of $m+1$ as $2$, $m+2$ as $3$ and so on) is $(N-m)c^m$. When $N$ is large we can ignore the $-m$. We then want $Nc^m\approx 1$ or $m\approx \frac{\log N }{-\log c}$ As this is the ratio of logs you can use any base you want. I would trust that the growth is logarithmic with $N$ but there could well be some constant of order $1$ multiplying the result.

Ross Millikan
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