This is a follow-up to an older question.
Let $r\in \mathbb{R}\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{|r-\frac{m}{n}|: m\in\mathbb{N}\}$$ be the best approximation of $r$ that can be obtained using $n$ as the denominator.
We inductively define $\text{appr}_r:\mathbb{N} \to \mathbb{N}$, the approximation sequence of $r$:
- $\text{appr}_r(0) = 1$, and
- $\text{appr}_r(n+1) = \min\{m\in\mathbb{N}: m \geq \text{appr}_r(n) \text{ and } \alpha_r(m) < \alpha_r(\text{appr}_r(n))\}$ for all $n\in \mathbb{N}$.
For $A\subseteq {\mathbb N}$ we say that $A$ is thin if $\lim\sup_{n\to\infty}\frac{|A\,\cap \,\{1,\ldots,n+1\}|}{n+1} = 0.$ We say that $r\in\mathbb{R}\setminus\mathbb{Q}$ is almost rational if the image of $\text{appr}_r$ is a thin subset of $\mathbb{N}$.
Question. Is there an almost rational $r\in\mathbb{R}\setminus\mathbb{Q}$?
