Timeline for Boundedness of sum of sin(sin(n))
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 7 at 21:41 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
links to Z&Z paper
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| Oct 7 at 15:08 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Zudlin —> Zudilin
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| Aug 6 at 17:54 | comment | added | David E Speyer | Some corrections: The irrationality measure of $\pi$ is known to be $\leq 7.11$, we don't know how much smaller it might be. All irrational numbers have irrationality measure $\geq 2$. The proof above works with any number with finite irrationality measure. A number with infinite irrationality measure is called a Liouville number en.wikipedia.org/wiki/Liouville_number . | |
| Aug 6 at 15:17 | comment | added | gnasher729 | So it seems that every irrational number has some constant telling us how irrational it is, pi has an irrational value 6.11, e might have a different value, and the sum in question is bounded if the value is greater than 2? | |
| Aug 6 at 12:25 | comment | added | David E Speyer | The earliest result I can find of this form is Mahler (1953), "On the approximation of $\pi$", who obtains $|\pi - p/q| > c q^{-42}$ and indicates that the exponent can be improved to $30$ but omits the details. ems.press/content/book-chapter-files/27418 The only earlier work that he cites is Feldman (1951) "The approximation of certain transcendental numbers. I. Approximation of logarithms of algebraic numbers" ; I wasn't able to extract a specific bound from Feldman. @PseudoNeo | |
| Aug 5 at 15:54 | comment | added | PseudoNeo | Seems that ZZ's bound is overkill for this theorem. How cheaper is it to prove a coarser bound, good enough to do the job? (Differently put: do I have a reasonable chance to make a problem out of this wonderful answer?) | |
| Aug 5 at 14:23 | comment | added | abx | If I understand correctly, the only information needed about $\pi$ is that its irrationality measure is finite. | |
| Aug 5 at 13:15 | vote | accept | Oleksandr Liubimov | ||
| Aug 5 at 13:08 | vote | accept | Oleksandr Liubimov | ||
| Aug 5 at 13:15 | |||||
| Aug 5 at 13:08 | comment | added | Oleksandr Liubimov | Mr. David, thank you very much for your solution, the theorems you stated are really nice and useful ! I did not know about them before. | |
| Aug 5 at 12:19 | comment | added | David E Speyer | Thanks! I've been wondering whether I can find a $\theta$ for which the sums of $\sin(\sin(n \theta))$ are unbounded. The tricky thing is that, if $\sin(\sin(\phi))$ is close to its maximum, then $\sin(\sin(3 \phi))$ is close to its minimum, so you tend to get a lot of cancellation. | |
| Aug 5 at 12:06 | comment | added | Dave Benson | This is a great answer. It shows how the ultimate point is that $\pi$ is not well approximated by rational numbers. | |
| Aug 5 at 12:05 | history | edited | GH from MO | CC BY-SA 4.0 |
edited body
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| Aug 5 at 11:49 | history | answered | David E Speyer | CC BY-SA 4.0 |