Romanoff theorem in a sparse set

Let A be any subset of positive integers, and P the set of all positive primes. Two of our results are: (a) the number of positive integers which are less than x and can be represented as 2 ^k + p (resp. p ? 2 ^k ) with k ∈ A and p ∈ P is more than 0.03A(log x/ log 2)π(x) for all sufficiently large x; (b) the number of positive integers which are less than x and can be represented as 2 ^q + p with p, q ∈ P is (1 + o(1)π(log x/ log 2)π(x). Four related open problems and one conjecture are posed.