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

Abstract

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 $1+1/4536$.

arXiv comment28 pages, 1 figure. v2: revised based on referee comments; results are unchanged