Timeline for Does every connected set that is not a line segment cross some dyadic square?
Current License: CC BY-SA 3.0
17 events
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| Mar 1 at 6:33 | comment | added | John Samples | Since you @mirko don't mind rephrasing-reposting the question I'd repost as a dimension/measure-theoretical question on curves in the plane, someone will spit some result at you really quick if you're convinced it's true. But I think it's false, you should be able to get good control of limiting behavior with a rectilinear construction and finish using basic results about direct limits of continua. | |
| Mar 1 at 6:23 | comment | added | John Samples | Really, some zero-span curve can't be constructed with just a uniform direct limit by oscillating off corners in Koch curve style with rectilinear paths by like a scaled factor of $> \frac{1}{4^n}$? In my head it works, the argument should go like 'what's the biggest dyadic square it crosses' and then induct. Has anyone tried to write it down? In my head it works. If there is no example then a dimension theorist probably can handwave why, you should tag dimension theory and GMT. | |
| S Jun 9, 2023 at 18:02 | history | bounty ended | CommunityBot | ||
| S Jun 9, 2023 at 18:02 | history | notice removed | CommunityBot | ||
| Jun 2, 2023 at 21:48 | comment | added | Mirko | @IanAgol No, I don't have a proof for the pseudoarc, though this is usually what I think of, as a totally path-disconnected continuum (i.e. path-components are points). If $C$ is say hereditarily indecomposable plane continuum I have considered (something elementary), take $\frac1n$-neighborhood (union of all $\frac1n$-balls with centers in $C$) this is path-connected, try to use it somehow (then let $n\to\infty$), but haven't had much success. Your comment reminded me that the pseudoarc is arc-like(chainable), so perhaps use this to modify the path-connected case, but I have to think about it | |
| Jun 2, 2023 at 3:25 | comment | added | Ian Agol | @Mirko: do you think that you can prove it for pseudo-arcs? | |
| S Jun 1, 2023 at 16:18 | history | bounty started | Mirko | ||
| S Jun 1, 2023 at 16:18 | history | notice added | Mirko | Draw attention | |
| Jun 1, 2023 at 15:50 | comment | added | Mirko | I believe I have a positive answer under the additional assumption that the set in question is path-connected. I posted a related question which eventually got a bit long, but see the definition of hsb plane continuum there, at the beginning, along with the Result after the "Edit May 29-31, 2023" here: mathoverflow.net/q/446092 . For completeness, here are a couple of other, related questions that I posted recently mathoverflow.net/q/446317 and math.stackexchange.com/q/4694709 . (I don't feel like posting my "path-connected" solution here, it is only a special case.) | |
| Feb 24, 2021 at 5:03 | comment | added | Anton Petrunin | Is it pure curiosity? In other words --- do you have some motivation for this question? | |
| Mar 30, 2014 at 20:02 | comment | added | Mirko | Hello Kevin, this is a nice question and seems difficult, though I came up with some ideas that I need to develop. Do you know if it is an open question, and if anyone has done work on it, any references? Did you find it somewhere in a book or a paper, or did you come up with it on your own? I am inclined to believe that there is no example, though I changed my mind about it several times yesterday and today. I will email a colleague whose area is closer to this question, to see if he knows anything or has any advice (just sending him the link to your question as posted here). Thank you, | |
| Feb 18, 2013 at 9:01 | comment | added | Kevin Johnson | Hi Gunter, I pointed out that there were two mistakes in this answer. A day later it was removed by somebody. | |
| Feb 17, 2013 at 13:17 | comment | added | Günter Rote | There used to be an interesting answer posted here. Where is it? Was it wrong? | |
| Feb 7, 2013 at 8:54 | comment | added | Kevin Johnson | Gunter Rote, I do not mean the condition $m\leq n$. E.g. the square $[1/8,5/8]\times [1/4,3/4]$ is a dyadic square with my definition. Perhaps you would like to call such a square a translated dyadic square. | |
| Jan 31, 2013 at 19:44 | comment | added | Marco Golla | I think that every path-connected component of such a non-crossing set should be a line segment. | |
| Jan 31, 2013 at 17:01 | comment | added | Günter Rote | I would have expected the condition $m\le n$. Is that what you mean? Or is this another version of the problem? Every line segment that is not $\pm45$ degrees, and every sufficiently smooth curve that is not a $\pm45$ degree line segment must cross a dyadic square (even with the strong definition of dyadic square). So if there are counterexamples they are pretty pathological. | |
| Jan 31, 2013 at 14:07 | history | asked | Kevin Johnson | CC BY-SA 3.0 |