We define \[ M(x,y) = \liminf_{q\to\infty} q||qx-y||. \] By Hurwitz Theorem, $M(x,0) \leq 1/5^{1/2}$ for all $x$ (and the value $1/5^{1/2}$ is achieved when $x$ is the golden number). The set of reciprocals of numbers $M(x,0)$ for all real $x$'s is called Lagrange spectrum, it is a heavily studied topic in the number theory. It is in particular known that the Lagrange spectrum contains all the numbers larger than so-called Freiman's constant, this part of Lagrange spectrum is called Hall's ray.

When we do not fix $y=0$ but allow its value to vary, we are talking about inhomogeneous Lagrange spectrum. In particular, it is known that if we fix any irrational $x$ and look at the possible values of $M(x,y)$, this set will contain some interval $[0,z]$.

We (Yann Bugeaud, Donghan Kim, Seonhee Lim, and myself) try to investigate the related, 'multifractal', question: given $x$ and $\alpha$, what is the set of $y$'s for which $M(x,y)$ equals $\alpha$? In the talk I will present our results.