Prime numbers with Beatty sequences

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes $p=q[\alpha n+\beta]+a$ with $n\le N$, where $\alpha,\beta$ are real numbers such that $\alpha$ is positive and irrational of finite type (which is true for almost all $\alpha$) and $a,q$ are integers with $0\le a<q\le N^\kappa$ and $\gcd(a,q)=1$, where $\kappa>0$ depends only on $\alpha$. We also prove a similar result for primes $p=[\alpha n+\beta]$ such that $p\equiv a\pmod q$.