There seems to be a lot of work on the upper bound for the [irrationality measure](http://mathworld.wolfram.com/IrrationalityMeasure.html) of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. [Looks like](https://mathoverflow.net/questions/134066/numbers-with-known-finite-irrationality-measure-greater-than-2?rq=1) it is the best one known, is it? What is the conventional wisdom on what $\mu(\pi)$ should be? Does a bound follow from some grand conjecture (Schanuel's, Riemann hypothesis, etc.)?


I am interested because Alekseyev [proved (Theorem 2)](http://arxiv.org/pdf/1104.5100v1.pdf) that $\frac{1}{n^u\sin{n}}$ converges to $0$ if $\mu(\pi)<u+1$ and diverges if $\mu(\pi)>u+1$. If we replace $\sin{n}$ by a sequence of independent random variables $\xi\_n$ uniformly distributed on $[-1,1]$ then $\frac{1}{n^u\xi\_n}$ converges to $0$ almost surely iff $\sum\frac{1}{n^u}<\infty$. Thus, if $\mu(\pi)>2$ then $\sin{n}$ is not 'uniformly distributed' on $[-1,1]$. On the other hand, if $\mu(\pi)=2$ then $\pi$ is just as irrational as $e$ and irrational algebraic numbers, while some people seem to [believe](https://mathoverflow.net/questions/53724/are-some-numbers-more-irrational-than-others) that it should be 'more irrational' than that.



Lower bound on the irrationality measure of $\pi$
The expected value is $2$, simply because almost all real numbers have irrationality measure $2$. Note that this is "almost all" in the sense of measure theory. In the sense of topological category, it's the other way around: the set of reals of measure $\infty$ is [residual](http://en.wikipedia.org/wiki/Meagre_set). As I [pointed out recently](https://mathoverflow.net/questions/176488/can-we-define-an-empirically-generic-real-number), it's a bit mysterious why measure theory makes better predictions than topological category for "naturally defined" numbers like $e$ or $\pi$, but experimentally, this appears to be the case, as [the upper bound](http://arxiv.org/abs/math/0404523) on the irrationality measure of $\pi$ is finite.


