Timeline for Measuring the quality of real approximation
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 11, 2022 at 1:47 | vote | accept | Dominic van der Zypen | ||
| Jul 10, 2022 at 21:38 | answer | added | user485571 | timeline score: 3 | |
| Jul 10, 2022 at 7:33 | comment | added | Gerry Myerson | For fixed $r$ and $c$, $c>2$, those approximations don't necessarily come from convergents to a continued fraction, but (as noted earlier) I suspect the $n$ that work are still few and far between. | |
| Jul 10, 2022 at 7:29 | comment | added | Dominic van der Zypen | That's very interesting, thanks @GerryMyerson! - Does your argument extend to $r$ irrational and $|r - (m/n)| < c(n^{-2})$ for $c > 2$? | |
| Jul 10, 2022 at 1:09 | comment | added | Gerry Myerson | Note that if $r$ is irrational, and $|r-(m/n)|<(2n^2)^{-1}$, then necessarily $m/n$ is a convergent of the continued fraction for $r$. For a given $r$, such $n$ form a very sparse set. I suspect that for any fixed $r$ and $c$ the $n$ for which $|r-(m/n)|<cn^{-2}$ will also be sparse. | |
| Jul 10, 2022 at 1:00 | history | edited | Dan Turetsky |
This question does not relate to computability theory, so I have removed that tag.
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| Jul 9, 2022 at 20:40 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |