Consecutive primes and Beatty sequences

Fix irrational numbers α,αˆ>1 of finite type and real numbers β,βˆ⩾0, and let B and Bˆ be the Beatty sequencesB:=(⌊αm+β⌋)m∈NandBˆ:=(⌊αˆm+βˆ⌋)m∈N. In this note, we study the distribution of pairs (p,p♯) of consecutive primes for which p∈B and p♯∈Bˆ. We conjecture that the estimate|{p⩽x:p∈B and p♯∈Bˆ}|=(ααˆ)−1π(x)+O(x(log⁡x)−3/2+ε) holds for every fixed ε>0, and we give a heuristic argument to support this prediction which relies (in part) on a strong form of the Hardy-Littlewood conjectures.