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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