Distribution Modulo One of αpγ + β for Special Classes of Primes

Let α,β∈R with α≠0, and let γ∈(0,5/6). Define the set M1 to consist of primes p such that p+2 is almost prime, and let M2 be the set of primes of the form p=a2+b2+1. We study the distribution of αpγ + β modulo one, as p ranges over the sets M1 and M2, respectively.