The irrationality measure of $\alpha\in\mathbb{R}$ is defined by $\displaystyle\mu(\alpha)=\inf\left\{\nu\in\mathbb{R}_+,\; \text{card}\left(\left\{\frac pq\in\mathbb{Q},\; 0<\left|\alpha-\frac pq\right|<\frac 1{q^\nu}\right\}\right)<+\infty\right\}$.
If $\mu(\alpha)=2$, it seems to me that $\inf$ is not $\min$ because of Dirichlet's theorem (which states that for every irrational number $\alpha$ there exist infinitely many integers $p,q$ such that $\displaystyle 0<\left|\alpha-\frac pq\right|<\frac 1{q^2}$).
My questions are :
- do you know some (irrational) numbers $\alpha$ for which $\mu(\alpha)\in[2,+\infty]$ is known to be a $\min$ ?
- do you know some (irrational) numbers $\alpha$ for which $\mu(\alpha)\in[2,+\infty]$ is known to be a $\inf$ but not a $\min$ ?
Many thanks.
