Timeline for Sequences of evenly-distributed points in a product of intervals
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2018 at 17:25 | vote | accept | Robin Saunders | ||
| Jul 12, 2018 at 15:41 | answer | added | Martin Roberts | timeline score: 3 | |
| Sep 8, 2010 at 5:52 | comment | added | Gerry Myerson | That's not what I meant, but, yes, those are meaningful questions. I think the Niederreiter paper looks at dispersion for $n\theta$ and finds it minimized for the golden mean. I don't know if he looks at dispersion for u. d. sequences, and I don't know if he looks at the minmax problem. But you now have several papers you can look at to see what results and what ideas you can try to apply to your questions. The Schilling paper is on Schilling's website. | |
| Sep 8, 2010 at 2:56 | comment | added | Robin Saunders | Well, it's still meaningful to ask about the minmax problem over the set of uniformly distributed or linear sequences, but the answer isn't covered by those papers. Is that what you meant? | |
| Sep 6, 2010 at 23:41 | comment | added | Gerry Myerson | @Robin, you may be able to get that paper from the author, Mark Schilling, who is still at Cal State Northridge. It does both the maxmin and minmax problems, and comes up with the same answer as the other two sources we've discussed here (good thing, that), the paper Helge found and the earlier work of Ruzsa cited by Niederreiter. But this answer is not, as Helge and I have noted, uniformly distributed, so you have to choose between the minmax property and the uniform distribution property - you can't have both. | |
| Sep 6, 2010 at 17:58 | comment | added | Robin Saunders | That's right, I'm interested for the moment in multiples of a fixed irrational. I would also be interested in any results about sequences which are not multiples of a fixed irrational, but which are uniformly distributed. In general, I'm interested in multiples of a fixed vector of irrationals. Sorry it took so long to make it clear what I meant; often I have a good intuitive idea of what problem I'm interested in, but it can take me a while to formalize it sufficiently to explain to others. | |
| Sep 6, 2010 at 12:41 | comment | added | Gerry Myerson | So, you're trying to minimize the maximum distance. Niederreiter and others have studied the dispersion, which is what you get when you maximize the minimum distance. Maybe there's some relation between the two. It also seems that you are only interested in sequences consisting of the multiples of a fixed irrational, even if there are other kinds of sequence with smaller maximum distance. It's still not clear to me which sequences in higher dimensions you are willing to consider. | |
| Sep 6, 2010 at 4:10 | comment | added | Robin Saunders | Take the first n points, and find the greatest distance between two adjacent points (for example; there are a number of other measures which also work). It will consistently be lower if the spacing is φ than for any other irrational. www.jstor.org/pss/2324121 looks relevant, although I don't currently have access to the full article. | |
| Sep 6, 2010 at 4:03 | comment | added | Gerry Myerson | @Helge, curiously, the $\log_2(2n-1)$ sequence in the arxiv paper turns up as the answer to mathoverflow.net/questions/31016/a-sequential-optimizing-task, where it is attributed to Ruzsa (which unfortunately gets spelled Rusza), although the exact reference isn't given (but it certainly predates the arxiv paper by many years). As for guessing what OP wants, I'm not sure I'm any better at that than you are. | |
| Sep 5, 2010 at 13:26 | comment | added | Helge | @Gerry: I am well aware that the sequence, I mentioned is not uniformly distributed. But it is DENSE in fact VERY DENSE! I mentioned this, because it gives another definition of what the original poster, might mean. (Which I have no clue about. I'm no good at guessing.) | |
| Sep 4, 2010 at 23:52 | comment | added | Gerry Myerson | I've read your edit, and I'm none the wiser as to how $\phi$ differs from all other irrationals, or how $n\phi$ differs from other uniformly distributed sequences of reals, or what you really want in higher dimensions. I suspect that if you ever get a precise formulation of "roughly-uniform distribution" you will find it's very like the "discrepancy" I've been writing about. Have you had a look at the Kuipers-Niederreiter book that Benoit and I have mentioned? | |
| Sep 4, 2010 at 20:27 | comment | added | Robin Saunders | Gerry: you're right, ordering of course won't work. I hadn't consciously thought about it, but I guess I pictured some measure based on taking the Voronoi diagram of the first n points, and then measuring distances between adjacent pairs of points. I think I'm thinking of something stronger than discrepancy. I know poorly-defined questions are anathema here, so I've tried above to give a clearer explanation of what I meant. | |
| Sep 4, 2010 at 20:27 | history | edited | Robin Saunders | CC BY-SA 2.5 |
clarify why φ is "special"
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| Sep 3, 2010 at 4:06 | answer | added | Gerry Myerson | timeline score: 2 | |
| Sep 2, 2010 at 23:13 | comment | added | Gerry Myerson | Robin, your question is unclear. I think the "stronger property" you attribute to multiples of $\phi$ has to do with uniformity of spacing between adjacent points (after ordering), but something like this happens for any irrational (look for The Three Gap Theorem). And you're interested in higher dimensions, but how do you propose to order a bunch of points up there? The usual way to measure how even a distribution is is via discrepancy, and there is a lot of work on low discrepancy sequences in high dimension, and the Kuiper-Niederreiter book will get you started. | |
| Sep 2, 2010 at 23:06 | comment | added | Gerry Myerson | @Helge, that paper only deals with $d=1$; Robin is asking about $d>1$. Also, that paper puts forward the sequence $u_n=\log_2(2n-1)$, which may suit the questions they discuss, but which isn't even uniformaly distributed modulo one, so is unlikely to be of interest here. | |
| Sep 2, 2010 at 16:31 | answer | added | dvitek | timeline score: 2 | |
| Sep 2, 2010 at 15:34 | comment | added | Helge | Do you mean a result like front.math.ucdavis.edu/0906.0045 ? | |
| Sep 2, 2010 at 15:30 | answer | added | Benoît Kloeckner | timeline score: 2 | |
| Sep 2, 2010 at 14:19 | comment | added | Robin Saunders | Yes, but this property is stronger than the sequence being uniformly distributed. | |
| Sep 2, 2010 at 14:02 | comment | added | Tony Huynh | Reminds me of Weyl's Equidistribution theorem. en.wikipedia.org/wiki/Equidistribution_theorem | |
| Sep 2, 2010 at 13:51 | history | asked | Robin Saunders | CC BY-SA 2.5 |