Timeline for Central limit theorem for irrational rotations
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Dec 6, 2023 at 13:03 | history | bounty ended | CommunityBot | ||
| S Dec 6, 2023 at 13:03 | history | notice removed | CommunityBot | ||
| Dec 1, 2023 at 21:38 | comment | added | Ronnie Pavlov | @Nikita This is not very helpful. Your answer came after another contradictory answer, and you're not saying what you think is wrong with it. Just to give some empirical evidence: your sum of real parts of powers of alpha is just a sum of cos(nx) (as you wrote). From Mathematica: Sum[Cos[kSqrt[2]],{k,10000}] = -0.939269... Sum[Cos[kSqrt[2]],{k,100000}] =-0.401409... Sum[Cos[k*Sqrt[2]],{k,1000000}] = 0.218388... It surely seems that these are not growing like n. (I'm aware that this is just one example, but try your favorite and see what happens!) | |
| Dec 1, 2023 at 21:11 | comment | added | Nikita Sidorov | @RonniePavlov see my answer | |
| Dec 1, 2023 at 15:01 | comment | added | Ronnie Pavlov | It's still possible there's just a huge miscommunication, but your sum is not on the order of n, it is bounded by Christophe's answer since it is (the real part of) a geometric series from a number of modulus 1. Can you explain why you don't seem to agree that this is true? | |
| Dec 1, 2023 at 14:10 | history | edited | Nikita Sidorov | CC BY-SA 4.0 |
answer
|
| Nov 30, 2023 at 13:34 | vote | accept | Nikita Sidorov | ||
| Nov 28, 2023 at 11:14 | history | edited | Nikita Sidorov | CC BY-SA 4.0 |
tidying up
|
| S Nov 28, 2023 at 11:09 | history | bounty started | Nikita Sidorov | ||
| S Nov 28, 2023 at 11:09 | history | notice added | Nikita Sidorov | Authoritative reference needed | |
| Nov 25, 2023 at 12:10 | history | edited | Nikita Sidorov | CC BY-SA 4.0 |
added 112 characters in body
|
| Nov 25, 2023 at 10:09 | comment | added | Nikita Sidorov | @GeraldEdgar Birkhoff's ergodic theorem works for all $x$ since $R_\alpha$ is uniquely ergodic. Does the CLT? | |
| Nov 24, 2023 at 21:30 | answer | added | Christophe Leuridan | timeline score: 7 | |
| Nov 24, 2023 at 13:29 | comment | added | Aleksei Kulikov | At the very least you need to exclude roots of unity. | |
| Nov 24, 2023 at 12:21 | comment | added | Gerald Edgar | So. Almost all $\alpha$ on the unit circle have this property, but do all algebraic integers on the unit circle have the property? It reminds me of the (open) question of whether algebraic irrational numbers must be normal. | |
| Nov 24, 2023 at 11:56 | comment | added | Christophe Leuridan | The sum is o(n) but it does not imply that the limit is 0. Example $\log(\sqrt{n})/\log(n) = 1/2$. Yet, in our example, the sums are bounded, and can be negative and the logarithm can be undefined. | |
| Nov 24, 2023 at 9:20 | history | edited | YCor |
edited tags; edited tags
|
|
| Nov 24, 2023 at 7:38 | history | asked | Nikita Sidorov | CC BY-SA 4.0 |