Timeline for The liminf of an expression involving an irrational rotation
Current License: CC BY-SA 4.0
9 events
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May 16, 2023 at 22:25 | comment | added | Nate River | @GHfromMO Thanks for the reference! Very interesting result. | |
May 16, 2023 at 22:12 | comment | added | GH from MO | There are countably many $a\in\mathbb{R}$ for which $\liminf n\{na\}$ exceeds $1/3$, and there are continuum many $a\in\mathbb{R}$ for which $\liminf n\{na\}$ equals $1/3$. See Cassels: An introduction to diophantine approximation (1957). Keyword: Markov spectrum (aka Markoff spectrum). | |
May 16, 2023 at 22:11 | history | edited | GH from MO |
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May 16, 2023 at 21:51 | comment | added | Asaf | @NateRiver a generic geodesic on $SL_{2}(\mathbb{R})/SL_{2}(\mathbb{Z})$ is equidistributed and in particular unbounded. For a number to be badly approximable, one needs the geodesic to be bounded. | |
May 16, 2023 at 21:38 | comment | added | Nate River | @AlekseiKulikov Right, please feel free to migrate the question. Do you think the corresponding question asking if it’s true for almost all irrationals is worth asking here? (I will make a new post, of course.) | |
May 16, 2023 at 21:24 | vote | accept | Nate River | ||
May 16, 2023 at 21:05 | answer | added | Iosif Pinelis | timeline score: 6 | |
May 16, 2023 at 20:58 | comment | added | Aleksei Kulikov | This is equivalent to the unboundedness of the terms in the continued fraction of $a$ (up to some technicalities), so in particular it is false for the quadratic irrationals such as $\phi$ or $\sqrt{2}$. But I think this question is more appropriate to math stackexchange. | |
May 16, 2023 at 20:26 | history | asked | Nate River | CC BY-SA 4.0 |