Distribution of α p 2 Modulo One with Prime Variable p of a Special Form

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that, for α ∈ ℝ / ℚ ,   β ∈ ℝ , and 0 < θ < 10 / 1561 , there exist infinitely many primes p , such that α p 2 + β < p − θ and p + 2 = P 4 , which constitutes an improvement upon the previous result.