Theoretical analysis proves a packing dimension variant of Keleti's conjecture in the plane, suggesting a direct resolution of the generalized Kakeya conjecture.
Let E ⊆ Rⁿ be a union of line segments and F ⊆ Rⁿ the set obtained from E by extending each line segment in E to a full line. Keleti's line segment extension conjecture posits that the Hausdorff dimension of F should equal that of E. Working in R², we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems.
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Bushling et al. (2024) studied this question.
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