Bounded Gaps between Primes of Special Form

Abstract

Let 0 < α, σ < 1 be arbitrary fixed constants, let $${{q}_{1}} < {{q}_{2}} < \ldots < {{q}_{n}} < {{q}_{{n + 1}}}$$ < … be the set of primes satisfying the condition $$\{ q_{n}^{\alpha }\} < \sigma $$ and indexed in ascending order, and let $$m \geqslant 1$$ be any fixed integer. Using an analogue of the Bombieri–Vinogradov theorem for the above set of primes, upper bounds are obtained for the constants c(m) such that the inequality $${{q}_{{n + m}}} - {{q}_{n}} \leqslant c\left( m \right)$$ holds for infinitely many n.