Some Diophantine Properties of the Sequence of S-units

Integers having no prime factors outside a fixed set of primes play important role and are heavily investigated in several parts of number theory. For example, they play special role in diophantine number theory; see e.g. the classical survey paper of Evertse, Győry, Stewart and Tijdeman [1] or Chapter 1 of the book of Shorey and Tijdeman [7] and the references given there. Further, the sequence formed of such integers is also of interest. To be precise, fix primes p1 < · · · < pt, and write sn for the sequence of integers composed of these primes, arranged in an increasing order. Tijdeman [8] and [9] provided sharp upper and lower bounds for the gaps between consecutive terms of the sequence, respectively. These bounds have the nice property that they are ”almost” equal. Namely, Tijdeman proved that