Timeline for Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$
Current License: CC BY-SA 4.0
40 events
| when toggle format | what | by | license | comment | |
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| S Jun 29, 2023 at 11:49 | history | bounty ended | Arbuja | ||
| S Jun 29, 2023 at 11:49 | history | notice removed | Arbuja | ||
| Jun 29, 2023 at 11:49 | vote | accept | Arbuja | ||
| Jun 29, 2023 at 3:28 | answer | added | Iosif Pinelis | timeline score: 2 | |
| Jun 29, 2023 at 3:05 | history | edited | Arbuja | CC BY-SA 4.0 |
Division by infinity is undefined. I fixed the issue. Also j/n should approach zero, not the other way around.
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| Jun 29, 2023 at 2:11 | history | edited | Arbuja | CC BY-SA 4.0 |
Redefining Uniformity by @IosofPinelis suggestions. Uniformity is not related to the Hausdorff dimension.
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| Jun 28, 2023 at 21:12 | answer | added | Iosif Pinelis | timeline score: 1 | |
| Jun 28, 2023 at 20:27 | comment | added | Arbuja | @IosifPinelis How would you suggest measuring the uniformity of measurable subsets of the unit square? Is there a paper on this? | |
| Jun 28, 2023 at 20:12 | comment | added | Iosif Pinelis | Previous comment continued: (iv) Overall, I think the idea of uniformity has hardly anything to do with the Hausdorff dimension (HD). E.g., I think almost everyone would agree that the entire unit square $S$ (of HD $=2$) is maximally uniform (in itself), whereas a fine uniform grid of $n^2$ points (of HD $=0$) in $S$ is approximately uniform in $S$ if $n$ is large. This, again, suggests that the HD has little (if anything) to do with the idea of uniformity. | |
| Jun 28, 2023 at 20:11 | comment | added | Iosif Pinelis | @DavisJohnson : Your definitions do make a bit more sense now, but they still do not make much sense to me. (i) Does such a set $A'$ as the one described in your post exist? (ii) Why is $A'$ not used at all in your conditions 1, 2, 3, or anywhere in your post? (iii) According to your point 1, if (say) $A$ is a small disk contained in the unit square (say $S$) but you choose(?) for some reason(?) $\alpha$ to be $<2$, then $A$ is maximally uniform in $S$ -- however small the disk $A$ is. In my mind, this does not correspond to any reasonable idea of uniformity. | |
| Jun 28, 2023 at 19:28 | comment | added | Arbuja | @AndyPutman I know you said no more edits but this one wasn't trivial. | |
| Jun 28, 2023 at 19:28 | comment | added | Arbuja | @IosifPinelis I think my definitions make sense now. | |
| Jun 28, 2023 at 19:27 | history | edited | Arbuja | CC BY-SA 4.0 |
Made necessary edits from @IosofPinelis suggestions
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| Jun 27, 2023 at 18:07 | comment | added | Iosif Pinelis | @DavisJohnson : This time I can parse your grammar. However, these definitions do not make much sense. In particular, for $\alpha=0$, taking $n=1$, we see that any nonempty subset $A$ of the unit square will be uniform in your sense. For $\alpha=2$, a subset $A$ of the unit square will be uniform in your sense iff its Hausdorff measure is $1$. | |
| Jun 27, 2023 at 17:44 | comment | added | Iosif Pinelis | @DavisJohnson : I have really tried to understand your posted question. But my (and your) efforts are not making anything clearer (at least, to me). In your latest response, the long sentence "If $n\ll j$ ..." is not even a sentence, and I cannot parse it even grammatically. In particular, what is "the ratio of all possible rectangles"? I guess I should just give up here. | |
| Jun 27, 2023 at 15:59 | comment | added | Iosif Pinelis | @DavisJohnson : (i) What do you mean by "'almost" all"? What do you mean by "measure has equivalent measures"? (ii) Do you only consider the small squares $[x_1,x_2]\times[x_1,x_2]$ near the diagonal of the unit square or all small squares all over the unit square? (iii) In your response (?) to my previous question (i), you did not even mention "w.r.t a subset uniform in $[0,1]\times[0,1]$". What subset? What do you mean by "a subset uniform in $[0,1]\times[0,1]$"? | |
| Jun 26, 2023 at 19:37 | comment | added | Iosif Pinelis | (i) What do you mean by "uniformity for $A$ w.r.t a subset uniform in $[0,1]\times[0,1]$"? In particular, what do you mean by "a subset uniform in $[0,1]\times[0,1]$"? (ii) In "$\alpha<\text{dim}_{\text{H}}(A)$", what is $\alpha$? | |
| S Jun 26, 2023 at 17:41 | history | bounty started | Arbuja | ||
| S Jun 26, 2023 at 17:41 | history | notice added | Arbuja | Authoritative reference needed | |
| Jun 22, 2023 at 22:06 | comment | added | Andy Putman | I really think the constant editing has to stop. Take this as a lesson for the future: don't post something until you have spent time polishing it into its final form. | |
| Jun 22, 2023 at 21:11 | comment | added | Arbuja | @AndyPutman I know I shouldn’t make edits but note that I don’t like the way the third bullet point of criteria 2. is written. If you say no, I won’t make an edit. | |
| Jun 22, 2023 at 17:29 | comment | added | Andy Putman | It is also the case that not all questions are going to get answers. This even includes well-received questions; for instance, I asked the following question 13 years ago and still don't know the answer (even though it is decently upvoted): mathoverflow.net/questions/38413/… | |
| Jun 22, 2023 at 17:27 | comment | added | Andy Putman | There are the general stackexchange rules, and there are also community norms. MO culture discourages excessive editing, and encourages you to be thoughtful before you post and get it right the first time (so as to not waste people's time). If you go and browse through well-received questions here, you'll see that they typically have at most one or two edits beyond the initial post. | |
| Jun 22, 2023 at 16:53 | comment | added | Arbuja | @AndyPutman I'll stop. I thought no one is responding because it's unclear. According to the website you can edit as long as there is a good reason to edit. | |
| Jun 22, 2023 at 16:52 | comment | added | Andy Putman | 13 edits is getting ridiculous. It gives off the vibe that you are using edits to try to keep this on the front page, which is not appropriate. | |
| Jun 22, 2023 at 16:39 | history | edited | Arbuja | CC BY-SA 4.0 |
I stated why I wanted to use gauge function h, the paper, or the other two posts
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| Jun 21, 2023 at 17:48 | history | edited | Arbuja | CC BY-SA 4.0 |
Making the post easier to read
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| Jun 21, 2023 at 15:47 | history | edited | Arbuja | CC BY-SA 4.0 |
Now that I deleted the answer by @DavidRoberts reccomendations, I will erase the last sentences
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| Jun 20, 2023 at 0:26 | comment | added | Arbuja | Got straight to the point using this post. | |
| Jun 20, 2023 at 0:25 | history | edited | Arbuja | CC BY-SA 4.0 |
Got straight to the point using @AlexRhea’s answer
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| Jun 16, 2023 at 20:56 | history | edited | Arbuja | CC BY-SA 4.0 |
the dimension is alpha, not d
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| Jun 16, 2023 at 16:59 | history | edited | Arbuja | CC BY-SA 4.0 |
Making the post easier to read.
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| Jun 15, 2023 at 22:18 | history | edited | Arbuja | CC BY-SA 4.0 |
The length of the square should be 1/n. Added "ideal" measure of uniformity.
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| Jun 12, 2023 at 14:45 | history | edited | Arbuja | CC BY-SA 4.0 |
See the comment below this post
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| Jun 12, 2023 at 14:42 | history | edited | Arbuja | CC BY-SA 4.0 |
deleted 4583 characters in body
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| Jun 11, 2023 at 16:09 | history | edited | Arbuja | CC BY-SA 4.0 |
Replaced R with A, made post easier to read
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| Jun 11, 2023 at 3:00 | history | edited | Arbuja | CC BY-SA 4.0 |
d should be alpha
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| Jun 11, 2023 at 2:53 | history | edited | Arbuja | CC BY-SA 4.0 |
Made the post easier to read
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| S Jun 11, 2023 at 2:48 | review | First questions | |||
| Jun 11, 2023 at 6:55 | |||||
| S Jun 11, 2023 at 2:48 | history | asked | Arbuja | CC BY-SA 4.0 |