  We use recent advances on the discretized sum-product problem to obtain new
bounds on the Hausdorff dimension of planar $(\alpha,2\alpha)$-Fursterberg
sets. This provides a quantitative improvement to the $2\alpha+\epsilon$ bound
of H\'era-Shmerkin-Yavicoli. In particular, we show that every
$1/2$-Furstenberg set has dimension at least $1 + 1/4536$.
