Timeline for Consequence of equidistribution or not?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Oct 14, 2010 at 3:41 | comment | added | Gerry Myerson | If $\theta$ has bounded partial quotients, then the discrepancy of the first $N$ terms of $n\theta\pmod1$ is $O(N^{-1}\log N)$. So for such $\theta$, $S_N=O(\log N)$. So it's probably not bounded for these $\theta$, but it is certainly less than any positive power of $N$. | |
| Oct 13, 2010 at 12:16 | comment | added | Gerry Myerson | I think it's bounded if the partial quotients of the irrational are bounded (as is the case for any quadratic irrational, for example). | |
| Oct 13, 2010 at 7:17 | comment | added | Dick Palais | I don't think so. For example the sequence 1/2,1/4,3/4,1/8,3/8,5/8, 1/16,3/16,... is equidistributed but not random. You need k-distributed for all (or at least for large k) for approximate pseudorandomness. I decided to test the sum $S_N$ when $\theta = \sqrt 2$ and in fact it seems to oscillate between -2 and 2 up to N = 1000. It never gets near 30. | |
| Oct 13, 2010 at 6:53 | comment | added | Fedor Petrov | @Gerry: It probably is adequate model (equidistributed sequences should behave as random sequences, natural to expect from them), but this fact is hard to prove in full generality. | |
| Oct 13, 2010 at 6:52 | comment | added | Dick Palais | You're right, it isn't---I was confusing equidistributed with pseudo-random. | |
| Oct 13, 2010 at 6:49 | history | undeleted | Dick Palais | ||
| Oct 13, 2010 at 6:47 | history | deleted | Dick Palais | ||
| Oct 13, 2010 at 5:08 | comment | added | Gerry Myerson | I don't think tosses of a fair coin is a good model for fractional part of multiples of an irrational. | |
| Oct 13, 2010 at 5:05 | history | answered | Dick Palais | CC BY-SA 2.5 |