  We use Maynard's methods to show that there are bounded gaps between primes
in the sequence $\{\lfloor n\alpha\rfloor\}$, where $\alpha$ is an irrational
number of finite type. In addition, given a superlinear function $f$ satisfying
some properties described by Leitmann, we show that for all $m$ there are
infinitely many bounded intervals containing $m$ primes and at least one
integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.
