The exact Hausdorff distance conjecture for the Markov and Lagrange spectra

Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively, with $L\subset M$. Define the Hausdorff distance by

$$d_H(L,M)=\sup_{m\in M\setminus L}d(L,m).$$

The constant $\delta_0$ is

$$\delta_0 = \frac{272052036746460995 - 3973474319367040 \sqrt{87} - 2762049221999040 \sqrt{18229} + 353887557067187 \sqrt{151905}}{226488036203921280} \approx 9.1094243388\times 10^{-8}.$$

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

The preceding theorem establishes that the Hausdorff distance is at least $\delta_0$. The conjecture asserts that no further portion of $M\setminus L$ lies farther from $L$; the surrounding discussion explains why longer odd non-semisymmetric words are expected to produce regions in smaller gaps, but no upper bound establishing equality is given.