On the multiplicative group generated by a dense sequence of integers (Q1084445)

Let \(A=\{a_ 1<a_ 2<...\}\) be a sequence of integers of upper density \(d>0\). It is shown that there is a set P of primes with a convergent sum of reciprocals and an integer \(k\leq 1/d\) such that every natural number n not divisible by any element of P is representable in the form \[ n=\prod^{s}_{i=1}a_{j_ i}^{\pm 1/k} \] for suitable s, \(j_ i\) and a suitable choice of the signs \(\pm \). This also follows from a result of the reviewer [Acta Arith. 32, 313-347 (1977; Zbl 0311.10048)].