On integers of the form $2^{k} \pm p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}} \cdots p_{r}^{\alpha _{r}}$

In this paper we prove that the set of positive odd integers which have no representation of the form $2^{n} \pm p^{\alpha } q^{\beta }$, where $p$, $q$ are distinct odd primes and $n, \alpha ,\beta$ are nonnegative integers, has positive lower asymptotic density in the set of all positive odd integers.