ON SETS OF INTEGERS WITH PRESCRIBED GAPS

Abstract

For a fixed set I of positive integers we consider the set of paths (P0,...,p(k) of arbitrary length satisfying p(l) - p(l-1) is-an-element-of I for l = 2,..., k and p0 = 1, p(k) = n. Equipping it with the uniform distribution, the random path length T(n) is studied. Asymptotic expansions of the moments of T(n) are derived and its asymptotic normality is proved. The step lengths p(l) - 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 p(k) are not specified. In the special case I = {m 1,m 2,...} (for some fixed m is-an-element-of 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.