  Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root
$\alpha$ and a real root $\beta$ of $\mathfrak{g}$, it is known that the
$\beta$-string through $\alpha$, denoted $R_\alpha(\beta)$, is finite. Given an
imaginary root $\beta$, we show that $R_\alpha(\beta)=\{\beta\}$ or
$R_\alpha(\beta)$ is infinite. If $(\beta,\beta)<0$, we also show that the
multiplicity of the root ${\alpha+n\beta}$ grows at least exponentially as
$n\to\infty$. If $(\beta,\beta)=(\alpha, \beta) = 0$, we show that
$R_\alpha(\beta)$ is bi-infinite and the multiplicities of $\alpha+n\beta$ are
bounded. If $(\beta,\beta)=0$ and $(\alpha, \beta) \neq 0$, we show that
$R_\alpha(\beta)$ is semi-infinite and the muliplicity of $\alpha+n\beta$ or
$\alpha-n\beta$ grows faster than every polynomial as $n\to\infty$. We also
prove that $\dim \mathfrak{g}_{\alpha+\beta} \geq \dim \mathfrak{g}_\alpha +
\dim \mathfrak{g}_\beta -1$ whenever $\alpha \neq \beta$ with $(\alpha,
\beta)<0$.
