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I know that $\tan n,n\in\mathbb{Z}_+$ is dense on $\mathbb{R}$, so there's no lower bound for $\tan n$.

But what if add a positive sequence $n^p,p\geq0,n\in\mathbb{Z}_+$ that increased faster than $n\bmod2\pi$ approaching $\dfrac{π}{2}$?

Where is the critical value of $p$?

Aster
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  • This problem seems directly connected to the Diophatine approximation of $\pi$. I think it's still open. – Cave Johnson Jun 15 '18 at 13:57
  • We don't know, the current upper bound for the irrationality measure for $\pi$ is around 7.6063. Any $p \gtrsim 6.6063$ will work. – achille hui Jun 15 '18 at 14:01
  • From the discussions here: https://math.stackexchange.com/questions/2820234/does-the-sequence-n-tann-n-in-mathbbn-have-a-lower-bound/2827850#comment5830875_2827850 , we have the following: \

    For $\pi/2=[a_1;a_2,a_3,\cdots ]$ a simple continued fraction,

    $$1. \mbox{The $2k$-th convergent $p_{2k}/q_{2k}$ has odd denominator.}$$

    $$\mbox{2. $a_{2k+1}\geq 2$, }$$

    Then $\tan p_{2k} < -1.2 p_{2k}$. Thus, if it is known that 1, 2 happen infinitely often, then $\tan n < -1.2 n$ infinitely often.

     – Sungjin Kim Jun 22 '18 at 13:07

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