Consecutive primes and Beatty sequences

William D. Banks, Victor Z. Guo

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:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{an…