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 \$sumlimits_{xin V(G)}f(x)\$of an independent broadcast \$f\$ on \$G\$.It is known that \$alpha(G)leq alpha_b(G)leq 4alpha(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 2alpha(G)\$provided that \$ggeq 6\$ and \$deltageq 3\$or that \$ggeq 4\$ and \$deltageq 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 2left(1-frac{1}{k}right)alpha(G)\$.Our results imply that lower bounds on the girth and the minimum degreeof a connected graph \$G\$can lower the fraction \$frac{alpha_b(G)}{alpha(G)}\$from \$4\$ below \$2\$, but not any further.]]> 0\$ and \$C = C(r) > 0\$ are constants depending on \$r\$.]]>