Timeline for Sequences of evenly-distributed points in a product of intervals
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 30, 2021 at 10:18 | comment | added | user76284 | @MartinRoberts Did you ever find a theoretical result along these lines? Or is it still an open problem? | |
| Sep 7, 2018 at 0:14 | comment | added | Martin Roberts | @RobinSaunders This is the next thing on my to-do list in terms of research articles. As you say, it will obviously provide a very firm theoretical foundation, but may just require time to sit down and think it all through carefully (or someone smarter than me!). If someone else in the math community wants to do it first, then go for it! | |
| Sep 6, 2018 at 17:58 | comment | added | Robin Saunders | This might be naive of me, but wouldn't it be an elementary (if perhaps tedious) exercise to prove that a sequence such as Martin's does in fact have low discrepancy? | |
| Sep 6, 2018 at 7:45 | comment | added | Kurisuto Asutora | It is worth noting that in the multi-dimensional setting d>1 no specific vector $\alpha$ is known for which $(n \alpha)$ is a low-discrepancy sequence. This is in stark contrast to the one-dimensional case. So the answer above provides some experimental facts, but the corresponding theoretic problem is still completely open. (This is essentially due to the fact that continued fractions are missing/problematic in the multi-dimensional case.) | |
| Jul 13, 2018 at 2:17 | history | edited | Martin Roberts | CC BY-SA 4.0 |
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| Jul 13, 2018 at 2:11 | history | edited | Martin Roberts | CC BY-SA 4.0 |
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| Jul 13, 2018 at 0:53 | comment | added | Gerry Myerson | (continued from previous comment) is far from being uniformly distributed, but its range is dense in the interval. And the theorem than the fractional part of $n\theta$ is uniformly distributed for irrational $\theta$ – I don't think anyone refers to this result as Weyl's criterion, which is something different altogether. | |
| Jul 13, 2018 at 0:50 | comment | added | Gerry Myerson | The first paragraph of this answer is very troubling. To write that, "in the limit as $n\to\infty$, all values of $x$ are equally likely," when what one really means by equidistribution is that, in the limit, all intervals of any given nonzero measure are equally likely, is surely a poor choice of words; almost all values of $x$ have probability zero, while the rest have probability one. Also, the equidistribution theorem says much more than that the range of the sequence is dense in the interval. The sequence where $x_n$ is the fractional part of $\log n$ (continued, next comment) | |
| Jul 12, 2018 at 23:38 | comment | added | Martin Roberts | Unfortunately not yet. This is still very much a work in progress! | |
| Jul 12, 2018 at 17:26 | comment | added | Robin Saunders | Thank you so much, this is great! Have you proved that this construction is optimal among Kronecker-type sequences? (I can certainly believe that it is.) | |
| Jul 12, 2018 at 17:25 | vote | accept | Robin Saunders | ||
| Jul 12, 2018 at 16:16 | history | edited | Martin Roberts | CC BY-SA 4.0 |
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| Jul 12, 2018 at 16:01 | review | Late answers | |||
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| Jul 12, 2018 at 15:53 | history | edited | Martin Roberts | CC BY-SA 4.0 |
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| Jul 12, 2018 at 15:46 | review | First posts | |||
| Jul 12, 2018 at 16:01 | |||||
| Jul 12, 2018 at 15:41 | history | answered | Martin Roberts | CC BY-SA 4.0 |