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nLab > Latest Changes: Kakeya Conjecture

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  1. Creating a New note about the Kakeya Conjecture. Just starting.

    Felipe Ponce

    v1, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeApr 9th 2020
    Hi, I cannot edit my note, I have been blocked by spam detector. How do I fix it? Thank you.
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2020

    Try breaking up the edit into a sequence of small edits. Sometimes that helps to deconfuse the spam filter.

  4. I have been trying to contribute repeatedly this entry, but until now I have been repeatedly blocked by the spam detector. This is my first entry, and I want to describe a few results about Kakeya conjecture. I do not plan to write an extensive treatise, just a few things. It is not easy to deal with the spam detector (thanks to Urs for helping me).

    Felipe Ponce

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2020
    • (edited Apr 10th 2020)

    Okay, in that case there is probly some text string in your edit that triggers the spam filter.

    You could try guessing which string that might be and try to remove it for the time being to see if that helps. I could imagine that colorful maths terminology such as “X-Ray transform” and similar has a non-vanishing chance of triggering the spam filter!

    Otherwise, only our admins can help: I have forwarded your issue, see there.

    • CommentRowNumber6.
    • CommentAuthorRodMcGuire
    • CommentTimeApr 10th 2020

    you could try posting what you are trying to add right here in the nForum so others might identify what is triggering the spam filter.

    Also others could see if they can add it.

    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 10th 2020
    • (edited Apr 10th 2020)

    Hi Felipe, apologies for the difficulties with the spam filter; there is a threshold for the permitted textual difference, and for new users, especially those without a corresponding nForum user, this threshold is tighter. For users with a significant number of edits and a corresponding nForum user, the threshold is to all intents and purposes non-existent, i.e. everything is permitted. As far as I see you were now able to post what you were trying to post; if not, let me know. As Urs wrote, breaking up a post is one way to get around it; or posting here and asking someone else to post for you as a last resort as Rod suggested (though posting yourself is better, to become a more ’trusted’ user).

    I’ll just re-iterate as I usually do that although there are occasional false negatives like this, the spam filter is catching numerous cases of spam as well (I checked again just now), and I think the benefits outweigh these unfortunate occasional cases.

    Many new users will begin a series of smaller edits rather than larger ones or new pages, and the spam filter should not kick in in those cases; the threshold is quite generous as long as there is already significant content on a page.

  8. Dear Felipe, interesting topic! I corrected a few typos. Further I have a few questions on the notation:

    • I don’t understand the notation L q(σ↦L x r(ℝ n−1))L^q(\sigma\mapsto L^r_x(\mathbb{R}^{n-1})).

    • Is Δ n\Delta^n some kind of higher order difference?

    • CommentRowNumber9.
    • CommentAuthorfelipe
    • CommentTimeJul 6th 2021

    Dear Daniel, I apologize for replying after so long, I truly didn’t see the message, I feel ashamed.

    The first notation refers to the L qL^q norm of a function σ↦f(σ)∈L r(ℝ n−1)\sigma\mapsto f(\sigma)\in L^r(\mathbb{R}^{n-1}). In this case σ↦Xf(σ,x)\sigma\mapsto Xf(\sigma,x), and for fixed σ\sigma we have Xf(σ,x)∈L x r(ℝ n−1)Xf(\sigma,x)\in L^r_x(\mathbb{R}^{n-1}); the xx is to emphasize.

    Δ N\Delta^N is a power of the laplacian.

    • CommentRowNumber10.
    • CommentAuthorporton
    • CommentTimeMar 6th 2025

    I add a reference to my proof sketch.

    diff, v16, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 6th 2025

    made cosmetic edits to wording and formatting of this entry

    diff, v17, current

    • CommentRowNumber12.
    • CommentAuthorporton
    • CommentTimeMar 6th 2025

    typo in a proper name (also the name of the conjecture).

    diff, v18, current

    • CommentRowNumber13.
    • CommentAuthorvictorporton
    • CommentTimeMar 8th 2025

    A Kakeya Set is a subset of a Euclidean space that contains a unit line segment for every direction. For example, a ball is a Kakeya set.

    The Kakeya Set Conjecture asserts that every compact Kakeya set E⊂ℝ nE\subset\mathbb{R}^n has Hausdorff dimension nn.

    Consider space with n=2n=2.

    A closed unit disk is a Kakeya set.

    Attack to the unit disk a two times smaller closed disk at the top, to that yet smaller two times at the top, etc. The resulting set conforming to the statement oft he conjecture is a fractal and therefore has <n&lt;n Hausdorff dimensionality. It is also a compact, because it is a closed bounded set.

    So, Kakeya conjecture is trivially false. Is it an error in nLab? What is the correct formulation?

    In Wikipedia, the requirement to be compact is missing but it has the same problem.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2025

    The Hausdorff dimension of a countable union of sets equals the supremum of the Hausdorff dimensions of the individual sets (e.g. here). Hence the Hausdorff dimension of your union of nn-disks is =n= n.

    Generally, the Hausdorff dimension is monotonic under inclusion of subsets (here) hence will never decrease even if attaching uncountable numbers of disks.

    • CommentRowNumber15.
    • CommentAuthorporton
    • CommentTimeMar 9th 2025

    proof sketch -> a little bit incomplete proof

    diff, v19, current

    • CommentRowNumber16.
    • CommentAuthorporton
    • CommentTimeMar 9th 2025

    updated proof URL

    diff, v19, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2025

    Care to comment on #14? It looks like in #13 you are unfamiliar with the basic ingredients of the Kakeya conjecture, but your reaction to this being pointed out is to claim a proof. I suggest you find another venue than the nLab to host your notes on this topic, for the time being.

    • CommentRowNumber18.
    • CommentAuthorporton
    • CommentTimeMar 10th 2025

    My proof of Kakeya can be considered complete (not a proof sketch or partial draft).

    diff, v20, current

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 11th 2025

    @Victor please have a read of https://terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/ . If your result were refereed I wouldn’t mind pointing to it, but the timing is curious, seeing as that you were never working in the area of this extremely challenging conjecture before.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 11th 2025
    • (edited Mar 11th 2025)

    Also, for prior art that has been refereed, see https://arxiv.org/abs/1704.07210 the burden is on you to explain, in your own space (i.e. not here, nor on the nLab page), why your approach supersedes all this technical work. Just ’using funcoids’ is not enough to overcome what people at the very top of mathematics struggled to achieve.

    I looked at your notes, and I’m not filled with confidence they do what you think.

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