On sets of integers with prescribed gaps |
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Authors: | Y Baryshnikov W Stadje |
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Institution: | (1) FB Mathematik/Informatik, Universität Osnabrück, Albrechtstrasse 28, D-49060 Osnabrück, Federal Republic of Germany |
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Abstract: | For a fixed setI of positive integers we consider the set of paths (p
o,...,p
k
) of arbitrary length satisfyingp
l
–p
l–1 I forl=2,...,k andp
0=1,p
k
=n. Equipping it with the uniform distribution, the random path lengthT
n
is studied. Asymptotic expansions of the moments ofT
n
are derived and its asymptotic normality is proved. The step lengthsp
l
–p
l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp
0 andp
k
are not specified. In the special caseI={m+1,m+2,...} (for some fixed m![isin](/content/r834gw3776174613/xxlarge8712.gif) ) 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 randomm-gap subset is also presented. |
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Keywords: | 05A16 05A17 60C05 |
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