I have a sequence $ a_{n}$, and the set

$ A=\{ m \le x | m \in \{a_{n}\}_{n=0}^{\infty} \} $ satisfying

$ \#A= x^\alpha+O(x^{\alpha-\varepsilon}). $

Also I have

$ \#\{ m \le x : p|m , m \in \{a_{n}\}_{n=0}^{\infty} \}= \frac{x^\alpha}{p}+O(x^{\alpha-\varepsilon}). $ for all p prime (of course those $p \ll x^{\varepsilon}$).

Is there any sieve(or maybe direct) theoritical approach from which I can conclude that R-smooth numbers in the above set has positive lower density(maybe asymptotic)?

Well I did try Buchstab's identity together with usual induction argument. I got no result.

Thanks