Hurwitz' theorem states that for every irrational $\xi$, there is infinitely many rationals $p/q$ such that

$$\left\vert \xi-\frac pq\right\vert<\frac 1{q^2\sqrt 5}.$$

The number $\sqrt 5$ is the best constant possible here, in the sense that if $C>\sqrt 5$, then there exist irrationals $x$ such that

$$\left\vert x-\frac pq\right\vert<\frac 1{q^2C}$$

is not satisfied for infinitely many rationals $p/q$.

Now, let's define for a real number $x$, $L(x)$ to be the biggest constant such that

$$\left\vert x-\frac pq\right\vert<\frac 1{q^2L(x)}$$

for infinitely many $p/q$.

As it is said here, an equivalent definition would be

$$L(x)=\left(\liminf_{q\to\infty} q^2\vert x-p/q\vert\right)^{—1}.$$

For example,

$$L\left(\frac {1+\sqrt 5}2\right)=\sqrt 5,\quad L(\sqrt 2)=2\sqrt 2.$$

The question.

I am interested in the set

$$L:=\{L(x),\ x\in \mathbb R\}$$

which is called the Lagrange Spectrum.

We already know that the part of $L$ lying in $[\sqrt 5,3)$ is discrete, and the final part lying in $[F,+\infty)$ where

$$F=\frac {2\,221\,564\,096+283\,748\sqrt{462}}{491\,993\,569}\approx 4.5278$$

is Freiman's constant, is continuous. We also know that the part lying between those two parts as a fractal structure.

Since I have found it nowhere, I am asking here:

Is there any graphical visualization of the set $L$? If not, would it be possible to make one?


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