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 \(2\alpha+\epsilon\) bound of Héra-Shmerkin-Yavicoli. In particular, we show that every \(1/2\)-Furstenberg set has dimension at least \(1 + 1/4536\).