On the distribution of α p²+β modulo one for primes p such that $p+2$ has no more two prime divisors

A classical problem in analytic number theory is to study the distribution of fractional part α pᵏ+β,\,k≥ 1 modulo 1, where α is irrational and p runs over the set of primes. For $k=2$ we consider the subsequence generated by the primes p such that $p+2$ is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.