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- W2891566379 abstract "We show that if $B subset mathbb{R}^n$ and $E subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $mathbb{R}^n$ such that every $P in E$ intersects $B$ in a set of Hausdorff dimension at least $alpha$ with $k-1 < alpha leq k$, then $dim B geq alpha +dim E/(k+1)$, where $dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $alpha$-Furstenberg set in the plane has Hausdorff dimension at least $alpha + 1/2$. More generally, we prove that if $B$ and $E$ are as above with $0 < alpha leq k$, then $dim B geq alpha +(dim E-(k-lceil alpha rceil)(n-k))/(lceil alpha rceil+1)$. We also show that this bound is sharp for some parameters. As a consequence, we prove that for any $1 leq k<n$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of $mathbb{R}^n$ has Hausdorff dimension at least $k+frac{s}{k+1}$." @default.
- W2891566379 created "2018-09-27" @default.
- W2891566379 creator A5087874914 @default.
- W2891566379 date "2018-09-12" @default.
- W2891566379 modified "2023-10-11" @default.
- W2891566379 title "Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces" @default.
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