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The following is still unknown:

Kakeya set conjecture: Define a Besicovitch set in $\mathbb{R}^n$ to be a set which contains a unit line segment in every direction. Is it true that such sets necessarily have Hausdorff dimension and Minkowski dimension equal to $n$?

Is the following known to be false?

My modified Kakeya set conjecture: Define a modified Besicovitch set in $\mathbb{R}^n$ to be a set which contains a line segment in every direction. Is it true that such sets necessarily have Hausdorff dimension and Minkowski dimension equal to $n$? (The requirement for the line segment to be unit is dropped, now line segments can be arbitrarily short: both absolutely and relatively each other.)

I ask, because it looks like for me, that I know how to prove the modified conjecture. Update: I don't know.

porton
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  • It seems that ChatGPT with Reason mode proved my conjecture false using Cantor set. I didn't yet check its proof. However, here it is: https://chatgpt.com/share/67c071ee-35f4-8001-b9c4-693a3b1c88a2 – porton Feb 27 '25 at 14:08
  • It looks like that ChatGPT mistakenly concluded that the closure of Cantor set is the entire line. Still need to check. – porton Feb 27 '25 at 20:09
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    (1) You should not trust AI programs with nontrivial math statements. (2) A proof of the Kakeya set conjecture in dimension 3 was recently posted on arXiv. – Moishe Kohan Feb 27 '25 at 20:43
  • As for your modified conjecture, I think it's equivalent to the original one. – Moishe Kohan Feb 28 '25 at 04:22
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    Tao discusses the arXiv preprint on dimension three at https://terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/ – Gerry Myerson Feb 28 '25 at 05:20
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    This has been proven in theorem 1.1 of this paper https://arxiv.org/abs/2203.15731 – Paul Feb 28 '25 at 05:43
  • @Paul: I think you should write this as an answer. Some people might find it useful. – Moishe Kohan Mar 03 '25 at 01:25

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The equivalence between the two conjectures has been proven in theorem 1.1 of Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements, by Tamás Keleti and András Máthé.

Jakobian
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Paul
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    "Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements", by Tamás Keleti and András Máthé. – Moishe Kohan Mar 05 '25 at 03:09