  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.
