Consecutive primes and Beatty sequences

Abstract

Fix irrational numbers $\alpha ,\hat{\alpha }>1$ of finite type and real numbers $\beta ,\hat{\beta }\ge 0$, and let $B$ and $\hat{B}$ be the Beatty sequences $$B:=(\lfloor \alpha m+\beta \rfloor {)}_{m\ge 1}\text{and}\hat{B}:=(\lfloor \hat{\alpha }m+\hat{\beta }\rfloor {)}_{m\ge 1}.$$ In this note, we study the distribution of pairs $(p,p^{♯})$ of consecutive primes for which $p\in B$ and $p^{♯}\in \hat{B}$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$|\{p\le x:p\in B\text{ and }{p}^{♯}\in \hat{B}\}|=(\alpha \hat{\alpha }{)}^{-1}\pi (x)+O(x(\log x{)}^{-3/2+\epsilon }),$$ where $\pi (x)$ is the prime counting function.

12 pages