Additive decomposability of multiplicatively defined sets

Let Q (T) denote the set of integers which are composed of prime factors from a given set of primes T only. Suppose that A+BQ′(T), where Q(T) and Q′(T) difier at flnitely many elements only. Also assume that τ log x+O(1). We prove that A(N)B(N) = O(N(log N)^2τ) holds. In the case we give an example where both A(N) and B(N) are of order of magnitude which shows that this is close to best possible.