Some Diophantine properties of the sequence of S-units

Let S   be a finite set of rational primes, and let snsn denote the increasing sequence of the positive integers having all their prime factors in S  . In this paper we develop a method to explicitly give the gaps in the sequence snsn. In other words, for any term snsn we can find both sn−1sn−1 and sn+1sn+1, at least in principle, without enumerating all terms of the sequence. In the case when S   contains two fixed primes, we even give an efficient algorithm to find these terms explicitly. Further, we apply our results to prove some Diophantine properties of the sequence snsn.