  We prove new bounds on the distributed fractional coloring problem in the
LOCAL model. Fractional $c$-colorings can be understood as multicolorings as
follows. For some natural numbers $p$ and $q$ such that $p/q\leq c$, each node
$v$ is assigned a set of at least $q$ colors from $\{1,\dots,p\}$ such that
adjacent nodes are assigned disjoint sets of colors. The minimum $c$ for which
a fractional $c$-coloring of a graph $G$ exists is called the fractional
chromatic number $\chi_f(G)$ of $G$.
  Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any
constant $\epsilon>0$, a fractional $(\Delta+\epsilon)$-coloring can be
computed in $\Delta^{O(\Delta)} + O(\Delta\cdot\log^* n)$ rounds. We show that
such a coloring can be computed in only $O(\log^2 \Delta)$ rounds, without any
dependency on $n$.
  We further show that in $O\big(\frac{\log n}{\epsilon}\big)$ rounds, it is
possible to compute a fractional $(1+\epsilon)\chi_f(G)$-coloring, even if the
fractional chromatic number $\chi_f(G)$ is not known. That is, this problem can
be approximated arbitrarily well by an efficient algorithm in the LOCAL model.
For the standard coloring problem, it is only known that an $O\big(\frac{\log
n}{\log\log n}\big)$-approximation can be computed in polylogarithmic time in
the LOCAL model. We also show that our distributed fractional coloring
approximation algorithm is best possible. We show that in trees, which have
fractional chromatic number $2$, computing a fractional $(2+\epsilon)$-coloring
requires at least $\Omega\big(\frac{\log n}{\epsilon}\big)$ rounds.
  We finally study fractional colorings of regular grids. In [Bousquet,
Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded
dimension, a fractional $(2+\epsilon)$-coloring can be computed in time
$O(\log^* n)$. We show that such a coloring can even be computed in $O(1)$
rounds in the LOCAL model.
