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

Daniel Di Benedetto, Joshua Zahl

Published 2021-12-15, updated 2022-11-07Version 2

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 $1 + 1/4536$.