On the packing dimension of Furstenberg sets

DOI10.1007/S11854-022-0203-XarXiv2006.15569MaRDI QIDQ6343900

Pablo Shmerkin

Publication date: 28 June 2020

Abstract: We prove that if $alphain(0,1/2]$, then the packing dimension of a set $Esubsetmathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $gealpha$ is at least $1/2+alpha+c(alpha)$ for some $c(alpha)>0$. In particular, this holds for $alpha$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $gealpha$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $alpha$-Furstenberg sets over the "trivial" estimate for all values of $alphain(0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $alpha$-Furstenberg set resembles a set of dimension close to $1/2+alpha$, if they exist, are rather sparse.

Mathematics Subject Classification ID

Length, area, volume, other geometric measure theory (28A75) Hausdorff and packing measures (28A78)

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