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New estimates on the size of $(α,2α)$-Furstenberg sets (2112.08249v2)
Published 15 Dec 2021 in math.CA, math.CO, and math.MG
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 H\'era-Shmerkin-Yavicoli. In particular, we show that every $1/2$-Furstenberg set has dimension at least $1 + 1/4536$.