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Do equal Hausdorff and packing dimension force full dimension for Kakeya sets in R^3?

Determine whether every Kakeya set K ⊂ R^3 with equal Hausdorff and packing dimension necessarily has Hausdorff dimension 3 (and hence packing dimension 3). This question seeks to understand if equality of Hausdorff and packing dimensions for Kakeya sets in three dimensions forces full fractal dimension.

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Background

The authors consider whether the natural condition of equal Hausdorff and packing dimensions for a Kakeya set in R3 implies full dimension. They note they cannot prove this implication; moreover, such a statement would entail that all Kakeya sets in R3 have packing dimension 3, itself a long-standing unresolved strengthening of the Kakeya conjecture.

To avoid trivial obstructions (e.g., adjoining an unrelated set X with equal dimensions to a Kakeya set K), the paper introduces a refined notion—"stably equal" Hausdorff and packing dimension—and proves that under this stability condition, Kakeya sets in R3 do have full Hausdorff and packing dimension.

References

We would like to say that every Kakeya set in R3 with equal Hausdorff and packing dimension must have dimension 3. Unfortunately, we cannot prove this statement, and indeed this would imply that every Kakeya set in R3 has packing dimension 3: let K ⊂ R3 be a Kakeya set, and let X ⊂ R3 be a compact set with equal Hausdorff and packing dimension dim_H(X) = dim_P(X) = dim_P(K). Then K = K ∪ X is a Kakeya set with equal Hausdorff and packing dimension, and this common value is dim_P(K).

— The Assouad dimension of Kakeya sets in $\mathbb{R}^3$ (2401.12337 - Wang et al., 22 Jan 2024) in Section 1 (Introduction), paragraph on equal Hausdorff and packing dimension (before Definition 1.3)