On sets of integers with prescribed gaps

Y. Baryshnikov, W. Stadje

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For a fixed set I of positive integers we consider the set of paths (po,..., pk) of arbitrary length satisfying pl-pl-1∈I for l=2,..., k and p0=1, pk=n. Equipping it with the uniform distribution, the random path length Tn is studied. Asymptotic expansions of the moments of Tn are derived and its asymptotic normality is proved. The step lengths pl-pl-1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values of p0 and pk are not specified. In the special case I={m+1, m+2,...} (for some fixed m∈ℕ) we derive the exact distribution of a random "m-gap" subset of {1,..., n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a random m-gap subset is also presented.

Original languageEnglish
Pages (from-to)83-98
Number of pages16
JournalMonatshefte für Mathematik
Issue number2
Publication statusPublished - Jun 1 1993
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


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