9 problems

Berstein's conjecture. The interval

Fix $\sigma\in \mathfrak S_3$, and let $k_1,k_2,k_3$ be nonnegative integers. Consider the generalized Markov tree $\mathrm{M}\mathbb T(k_1,k_2,k_3,\sigma)$. For any irreducible fr…

Fix $\sigma\in \mathfrak S_3$ and consider the Markov tree $\mathrm{M}\mathbb T(0,0,0,\sigma)$. For any irreducible fractions $t,s\in[0,1]$, let $n_t$ and $n_s$ be Markov numbers w…

Let $\mathcal M_{0,0,0}$ denote the classical discrete Markov spectrum, and let $\mathcal L(\alpha)$ denote the Markov-Lagrange value associated with an irrational $\alpha$. Two ir…

Exact Hausdorff distance conjecture. The Hausdorff distance between $M$ and $L$ is precisely $\delta_0$.

Let $L$ be the classical Lagrange spectrum, let $t_1$ be the first transition point, and let $delta>0$. The first transition point is defined by … where…

Post-transition interior conjecture. For every $\epsilon>0$,

Let $\mathcal{M}_\infty$ be the Markov spectrum of modular billiards, consisting of the values $\lambda_\infty(\mathcal{B})$ obtained from modular billiards $\mathcal{B}$. For each…

Moreira's question. For every $\delta>0$, one has