Primes in Intersections of Beatty Sequences

In this note we consider the question of whether there are infinitely many primes in the intersection of two or more Beatty sequences ⌊ ξjn + ηj⌋, n ∈ N, j = 1,...,k. We begin with a straightforward sufficient condition for a set of Beatty sequences to contain infinitely many primes in their intersection. We then consider two sequences when one ξj is rational. However, the main result we establish concerns the intersection of two Beatty sequences with irrational ξj. We show that, subject to a natural "compatibility" condition, if the intersection contains more than one element, then it contains infinitely many primes. Finally, we supply a definitive answer when the compatibility condition fails.