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

Daniel Di Benedetto; Joshua Zahl

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

Classical Analysis and ODEs 2022-11-09 v2 Combinatorics Metric Geometry

Authors: Daniel Di Benedetto , Joshua Zahl

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 $1 + 1/4536$ .

Keywords

furstenberg sets hausdorff dimension sum-product problem discretized estimates fractal geometry

Cite

@article{arxiv.2112.08249,
  title  = {New estimates on the size of $(\alpha,2\alpha)$-Furstenberg sets},
  author = {Daniel Di Benedetto and Joshua Zahl},
  journal= {arXiv preprint arXiv:2112.08249},
  year   = {2022}
}

Comments

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

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

Abstract

Keywords

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