  The spectral set conjecture, also known as the Fuglede conjecture, asserts
that every bounded spectral set is a tile and vice versa. While this conjecture
remains open on ${\mathbb R}^1$, there are many results in the literature that
discuss the relations among various forms of the Fuglede conjecture on
${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly
stronger universal tiling (spectrum) conjectures on the respective groups. In
this paper, we clarify the equivalences between these statements in dimension
one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is
true, then every spectral set with rational measure must have a rational
spectrum. We then investigate the Coven-Meyerowitz property for finite sets of
integers, introduced in \cite{CoMe99}, and we show that if the spectral sets
and the tiles in ${\mathbb Z}$ satisfy the Coven-Meyerowitz property, then both
sides of the Fuglede conjecture on ${\mathbb R}^1$ are true.
