For a given real number $x$, let $R_x$ be the set of real numbers $r$ such that the inequality
$$\displaystyle \left| x - \frac{p}{q} \right| < \frac{1}{q^s}$$ frac{1}{q^r}$$has at most finitely many solutions with integers p,q. Define the irrationality measure of x, say \mu(x), to be the infimum of R_x. It is known that if x is algebraic and not rational, then \mu(x) is 2, by Roth's Theorem. It is trivial that if x is rational, then \mu(x) = 1. I believe it is also known that all real numbers except a set of measure 0 has irrationality measure of 2, but I am unsure of the reference. For some known transcendental numbers, upper bounds for \mu are known. For example, we know that \mu(\pi) < 7.6063 (Salikhov, V. Kh. "On the Irrationality Measure of ." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008.) Are there any general results concerning a set of transcendental numbers x with \mu(x) = 2? Are there any known, 'interesting' numbers (expressible in well-known functions or constants) x with \mu(x) = 2? 1 # Numbers with known irrationality measures? For a given real number x, let R_x be the set of real numbers r such that the inequality$$\displaystyle \left| x - \frac{p}{q} \right| < \frac{1}{q^s} has at most finitely many solutions with integers $p,q$. Define the irrationality measure of $x$, say $\mu(x)$, to be the infimum of $R_x$.
It is known that if $x$ is algebraic and not rational, then $\mu(x)$ is 2, by Roth's Theorem. It is trivial that if $x$ is rational, then $\mu(x) = 1$. I believe it is also known that all real numbers except a set of measure 0 has irrationality measure of 2, but I am unsure of the reference.
For some known transcendental numbers, upper bounds for $\mu$ are known. For example, we know that $\mu(\pi) < 7.6063$ (Salikhov, V. Kh. "On the Irrationality Measure of ." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008.)
Are there any general results concerning a set of transcendental numbers $x$ with $\mu(x) = 2$? Are there any known, 'interesting' numbers (expressible in well-known functions or constants) $x$ with $\mu(x) = 2$?