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May 17, 2023 at 15:19 history edited Iosif Pinelis CC BY-SA 4.0
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May 17, 2023 at 0:55 comment added Oleg Eroshkin @mathworker21 It's true that this property holds for almost all real. Khinchin proved in 1926, that if $\psi(x)$ is a positive decreasing real-valued function such that $\sum \psi(n)$ diverges, then for almost all real $\theta$ the inequality $\{q\theta\}<\psi(q)$ has infinitely many solutions. See Cassels's book mentioned by GH, Chapter 7.
May 17, 2023 at 0:09 comment added mathworker21 @NateRiver Ok, I was wrong. Was thinking of the reverse. Look up the proof of Duffin-Schaeffer by Koukoulopoulos-Maynard for background.
May 16, 2023 at 23:33 comment added Nate River @mathworker21 Would you mind providing a quick hint? I thought of using the result that almost all reals have irrationality measure $2$, but it seems like this on its own is not sufficient to prove the result…
May 16, 2023 at 23:03 comment added mathworker21 @NateRiver Yes, it should be an exercise to show it holds for almost all irrationals (equivalently, almost all reals).
May 16, 2023 at 22:13 comment added GH from MO @NateRiver See my comment below the original post.
May 16, 2023 at 21:24 vote accept Nate River
May 16, 2023 at 21:24 comment added Nate River Ahh, thank you. It is clear now. I wonder if this at least holds for almost all irrationals, since this property seems specific to quadratic irrationals as mentioned.
May 16, 2023 at 21:20 comment added mathworker21 @NateRiver See, e.g., my comment here.
May 16, 2023 at 21:10 history edited Iosif Pinelis CC BY-SA 4.0
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May 16, 2023 at 21:08 comment added Nate River Sorry, why is $q$ times the infimum bounded below? Is this to do with the irrationality measure being $2$?
May 16, 2023 at 21:05 history answered Iosif Pinelis CC BY-SA 4.0