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On the packing dimension of Furstenberg sets


We prove that if α ∈ (0, 1/2], then the packing dimension of a set E ⊂ ℝ2 for which there exists a set of lines of dimension 1 intersecting E in Hausdorff dimension ≥ α is at least 1/2 + α + c(α)for some c(α) > 0. In particular, this holds for α-Furstenberg sets, that is, sets having intersection of Hausdorff dimension ≥ ≥ 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 α-Furstenberg sets over the “trivial” estimate for all values of α ∈ (0, 1). The proof extends to more general families of lines, and shows that the scales at which an α-Furstenberg set resembles a set of dimension close to 1/2 + α, if they exist, are rather sparse.

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Correspondence to Pablo Shmerkin.

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P. S. was partially supported by Project PICT 2015-3675 (ANPCyT)

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Shmerkin, P. On the packing dimension of Furstenberg sets. JAMA 146, 351–364 (2022).

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