Timeline for Sequences similar to $\{n\alpha\}$ that are both equidistributed and truly random-like
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Dec 4, 2020 at 22:37 | answer | added | Goldstern | timeline score: 1 | |
| Nov 30, 2020 at 15:20 | vote | accept | Vincent Granville | ||
| Nov 30, 2020 at 1:12 | answer | added | Yuval Peres | timeline score: 6 | |
| Nov 29, 2020 at 21:12 | comment | added | Vincent Granville | @Goldstern: I think you should put all your comments into an answer. It has more than I was expecting to accept an answer. You may want to wait a little bit in case someone comes up with some other interesting thing, yet your comments are highly valuable (if you could add some references, it would be perfect). | |
| Nov 29, 2020 at 19:52 | history | edited | Vincent Granville | CC BY-SA 4.0 |
added 18 characters in body
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| Nov 29, 2020 at 19:31 | history | edited | Vincent Granville | CC BY-SA 4.0 |
edited body
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| Nov 29, 2020 at 19:29 | comment | added | Vincent Granville | @Gerry: I am a little familiar (not much) with low-discrepancy sequences. It looks like the goal with these sequences is to achieve less randomness rather than more, while at the same time not leaving big holes in the space covered by $n$-uples $(x_n,\dots,x_{n+k})$. This is because it works best in numerical analysis problems. | |
| Nov 29, 2020 at 19:25 | history | edited | Vincent Granville | CC BY-SA 4.0 |
See update at the bottom (added scatterplot, definition of random-like, and more)
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| Nov 29, 2020 at 17:18 | comment | added | Vincent Granville | Thank you for all the very useful comments. | |
| Nov 29, 2020 at 12:51 | comment | added | Goldstern | Remark 2: If you consider the sequence $\alpha^n$ mod 1 (exponential instead of polynomial), then for almost all values $\alpha>1$ you will get a sequence $(x_n)$ such that for all $k$, the sequence $(x_{n},\ldots,x_{n+k})$ is equidistributed in $[0,1]^{k+1}$. | |
| Nov 29, 2020 at 12:48 | comment | added | Goldstern | Remark 1: pairs $(x_n,x_{n+1}) $ of successive values in the sequence $x_n=n\alpha$ (mod 1, of course) lie on a straight line, as shown in your plot. Similarly, triples $(x_n,x_{n+1},x_{n+2})$ of successive values lie on a plane if $x_n= n^2\alpha$, and a similar linear dependence exists for higher integer values of the exponent $p$ in $n^p\alpha$. | |
| Nov 29, 2020 at 11:33 | comment | added | Gerry Myerson | Are you familiar with Halton sequences, Vincent? | |
| Nov 29, 2020 at 10:13 | comment | added | YCor | One might require that for every $k$, every nontrivial linear combination of $x_{n+1},\dots,x_{n+k}$ is equidistributed modulo 1? | |
| Nov 29, 2020 at 9:37 | comment | added | Emil Jeřábek | No specific sequence is “truly random”. | |
| Nov 29, 2020 at 7:34 | history | edited | Vincent Granville | CC BY-SA 4.0 |
added 70 characters in body
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| Nov 29, 2020 at 7:28 | history | asked | Vincent Granville | CC BY-SA 4.0 |