Numbers with known finite irrationality measure greater than 2 For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$ be defined as the supremum of all real numbers $\mu$ such that 
$$ \left| \alpha-\frac{p}{q}\right| \le \frac{1}{q^\mu} $$ has infinitely many solutions $\frac{p}{q} \in \mathbb{Q}$, where $q \ge 1$.
It's known that if $\alpha$ is rational, then $\mu(\alpha)=1$ and if $\alpha$ is algebraic and irrational, then $\mu(\alpha)=2$ by Roth's Theorem. The set of all $\alpha$ such that $\mu(\alpha) >2$ has Lebesgue measure 0 by Khinchin's Theorem.
One can explicitly write down real numbers $\alpha$ such that $\mu(\alpha) = \infty$ (e.g. Liouville's constant) and transcendental real numbers $\alpha$ such that $\mu(\alpha)=2$ (e.g. $\alpha =e$, see also this thread). It's also possible to find upper bounds on $\mu(\alpha)$ for some real numbers such as $\pi$. But I don't know of a single example of a real number $\alpha$ whose irrationality measure is known, finite and greater than 2.
Question: Is there an example of a real number whose irrationality measure $\mu(\alpha)$ is known exactly and satisfies $2<\mu(\alpha)<\infty$?
 A: The irrationality measure of the Champernowne constant $C_b$ in base $b>2$ is exactly $b$.
A: Irrationality measure is a question about approximation by rationals. The continued fraction expansion gives the best approximations and controls their quality. Irrationality measure is a kind of asymptotic growth of the continued fraction expansion. Asking about the irrationality measure of a particular number is asking properties of its continued fraction expansion. But if you are willing to specify numbers by their continued fraction expansion, it is easy to write down a continued fraction expansion with the desired measure. Inductively define the continued fraction $[a_1,a_2,\ldots]$ by setting $a_{n+1}=\lceil q_n^{\mu-1}\rceil$, where the convergent is $\frac{p_n}{q_n}$. Then $q_{n+1}=a_{n+1}q_n+q_{n-1}$, so the error $(q_nq_{n+1})^{-1}$ is about $q_n^{-\mu}$.
It's a bit of a cop-out, but it's definitely worth mentioning.
A: The answer is yes - see for example



Yann Bugeaud Diophantine approximation and Cantor sets Math. Ann. (2008) 341:677–684



