TITLE: Irrationality measure of arctan(1/3)
QUESTION [8 upvotes]: I recently came across the concept of the irrationality measure. It really fascinated me and when I was looking for known values $\mu(x)$ for mathematical constants $x$, I also came across this paper: On the irrationality measure of arctan 1/3
Unfortunately it could't help me out quite well.
What is the irrationality measure of $\arctan(1/3)$? Are there upper and lower bounds? Which famous constant have known irrationality measures or upper/lower bounds? (Except for the ones, that can easily be found on mathworld e.g. $\pi, \ln(2),\ln(3), \pi^2, \zeta(3)$)?

REPLY [8 votes]: As mentioned by GH from MO, the irrationality measure
for almost all real numbers is 2. However,
computing it for a particular number 
is a notoriously difficult problem. For an irrational algebraic number this measure is indeed 2, but this is a pretty hard theorem by Roth for which he got a Fields medal. 
Other then that, to my knowledge the only "famous constant" with the known irrationality measure is the number $e$. It is, again, equal to 2 and this is a very old result essentially known to Euler. (It follows from the expansion of $e$ into a continued fraction.) 
For other "famous", or even not that famous,  constants only upper bounds are known.
[EDIT] There is actually a pretty good answer here on MO which I missed somehow:
Numbers with known irrationality measures?