Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$

Abstract

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus \mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence $\{a_r{\}}_{r\in \mathbb{N}}$ converging to $3$, such that for any $r$ one can find a subset ${\mathcal{B}}_r\subset (a_{r+1},a_r)\cap {\mathcal{L}}^{{}^{\prime }}$ with the property that the Hausdorff dimension of $((a_{r+1},a_r)\cap \mathcal{L})\setminus {\mathcal{B}}_r$ is less than the Hausdorff dimension of ${\mathcal{B}}_r$ and for $t\in {\mathcal{B}}_r$ the sets of irrational numbers with Lagrange value bounded by $t$ and exactly $t$ respectively, have the same Hausdorff dimension. We also show that, as $t$ varies in ${\mathcal{B}}_r$, this Hausdorff dimension is a strictly increasing function. Finally, in relation to $\mathcal{M}\setminus \mathcal{L}$, we find $C>0$ such that we can bound from above the Hausdorff dimension of $(\mathcal{M}\setminus \mathcal{L})\cap (-\infty ,3+\rho )$ by $\frac{\log (\abs{\log ρ})-\log (\log(\abs{\log ρ}))+C}{\abs{\log ρ}}$ if $\rho >0$ is small.