  An independent broadcast on a connected graph $G$ is a function $f:V(G)\to
\mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at
most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for
every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast
independence number $\alpha_b(G)$ of $G$ is the largest weight
$\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$.
  It is known that $\alpha(G)\leq \alpha_b(G)\leq 4\alpha(G)$ for every
connected graph $G$, where $\alpha(G)$ is the independence number of $G$. If
$G$ has girth $g$ and minimum degree $\delta$, we show that $\alpha_b(G)\leq
2\alpha(G)$ provided that $g\geq 6$ and $\delta\geq 3$ or that $g\geq 4$ and
$\delta\geq 5$. Furthermore, we show that, for every positive integer $k$,
there is a connected graph $G$ of girth at least $k$ and minimum degree at
least $k$ such that $\alpha_b(G)\geq 2\left(1-\frac{1}{k}\right)\alpha(G)$. Our
results imply that lower bounds on the girth and the minimum degree of a
connected graph $G$ can lower the fraction $\frac{\alpha_b(G)}{\alpha(G)}$ from
$4$ below $2$, but not any further.
