An improved bound for the dimension of $(\alpha ,2\alpha )$-Furstenberg sets

We show that given $\alpha \in (0,1)$ there is a constant $c=c(\alpha )>0$ such that any planar $(\alpha ,2\alpha )$-Furstenberg set has Hausdorff dimension at least $2\alpha +c$. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps.