The dimension of intersection of Lagrange spectrum with a half-line may assume any value in [0,1]

You need to know: Notations: ${\mathbb R}$ for the set of real numbers, ${\mathbb Q}$ for the set of rational numbers, ${\mathbb R}\setminus {\mathbb Q}$ for the set of irrational numbers, $\text{dim}(A)$ for the Hausdorff dimension of set $A \subseteq {\mathbb R}$ .

Background: For $\alpha \in {\mathbb R}\setminus {\mathbb Q}$ , let $k(\alpha)$ be the supremum of all $k>0$ , for which the inequality $\left|\alpha-\frac{p}{q}\right|<\frac{1}{kq^2}$ holds for infinitely many rational numbers $\frac{p}{q}$ . The Lagrange spectrum is the set $L= \{k(\alpha)\,|\,\alpha\in {\mathbb R}\setminus {\mathbb Q}, k(\alpha) <+\infty\}$ of all possible finite values of $k(\alpha)$ . For any $t \in {\mathbb R}$ , let $d(t) = \text{dim}(L \cap (-\infty, t))$ . A function $f:{\mathbb R}\to[0,1]$ is called surjective if for every $x\in[0,1]$ there exists $t\in {\mathbb R}$ such that $f(t)=x$ .

The Theorem: On 17th December 2016, Carlos Moreira submitted to arxiv a paper in which he proved that $d(t)$ is a continuous non-decreasing surjective function from ${\mathbb R}$ to $[0,1]$ , such that $\max\{t\in {\mathbb R}\,|\,d(t)=0\}=3$ and $d(\sqrt{12}-\delta)=1$ for some $\delta>0$ .

Short context: Approximating irrational numbers by rationals is an old topic in number theory. In 1891, Hurwitz proved that every irrational number $\alpha$ can be approximated by infinitely many rational numbers $\frac{p}{q}$ with accuracy $\left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$ . The constant $\sqrt{5}$ in this Theorem is the best possible which works for all $\alpha$ . Function $k(\alpha)$ defined above is the best constant which works for any specific $\alpha$ , and it measures “how well” $\alpha$ can be approximated by rationals. Hurwitz Theorem states that $k(\alpha)\geq \sqrt{5}$ for all $\alpha \in {\mathbb R}\setminus {\mathbb Q}$ . This is the best possible because $k\left(\frac{1+\sqrt{5}}{2}\right)=\sqrt{5}$ . In terms of Lagrange spectrum L, this means that $\sqrt{5}$ is the smallest element of L. Properties of L are critical to understand rational approximation, and the Theorem (which answers a question asked by Bugeaud in 2008) deeply enrich our understanding of L.