ON SUMS OF PRIMES FROM BEATTY SEQUENCES

Let k ≥ 2 and α1,β1, . . . ,αk,βk be reals such that the αi’s are irrational and greater than 1. Suppose further that some ratio αi/αj is irrational. We study the representations of an integer n in the form p1 + p2 + · · · + pk = n, where pi is a prime from the Beatty sequence Bi = {n ∈ N : n = [αim + βi] for some m ∈ Z} .