  The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r)
is a Beatty set with density p/q. We derive a formula for the Fourier transform
\hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 \pi i j \floor{nq/p+r} / q}. A. S.
Fraenkel conjectured that there is essentially one way to partition the
integers into m>2 Beatty sets with distinct densities. We conjecture a
generalization of this, and use Fourier methods to prove several special cases
of our generalized conjecture.
