New estimates on the size of $(\alpha ,2\alpha )$-Furstenberg sets

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 $2\alpha +ϵ$ bound of H'era-Shmerkin-Yavicoli. In particular, we show that every $1\mathrm{/}2$-Furstenberg set has dimension at least $1+1\mathrm{/}4536$.