Fractal dimensions of the Markov and Lagrange spectra near $3$

Abstract: The Lagrange spectrum $\mathcal{L}$ and Markov spectrum $\mathcal{M}$ are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. It is known that the Hausdorff dimension of the intersection of these sets with any half-line coincide, that is, $\mathrm{dim}{\mathrm{H}}(\mathcal{L} \cap (-\infty, t)) = \mathrm{dim}{\mathrm{H}}(\mathcal{M} \cap (-\infty, t)):= d(t)$ for every $t \geq 0$. It is also known that $d(3)=0$ and $d(3+\varepsilon)>0$ for every $\varepsilon>0$. We show that, for sufficiently small values of $\varepsilon > 0$, one has the approximation $d(3+\varepsilon) = 2\cdot\frac{W(e^{c_0}|\log \varepsilon|)}{|\log \varepsilon|}+\mathrm{O}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)$, where $W$ denotes the Lambert function (the inverse of $f(x)=xe^x$) and $c_0=-\log\log((3+\sqrt{5})/2) \approx 0.0383$. We also show that this result is optimal for the approximation of $d(3+\varepsilon)$ by "reasonable" functions, in the sense that, if $F(t)$ is a $C^2$ function such that $d(3+\varepsilon) = F(\varepsilon) + \mathrm{o}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)$, then its second derivative $F''(t)$ changes sign infinitely many times as $t$ approaches $0$.