On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

Let L and M denote the classical Lagrange and Markov spectra, respectively. It is known that L M and that M L. Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function d₋₎₂ (t) that gives the local Hausdorff dimension at a point t of L'. Specifically, we construct several intervals (having non-trivial intersection with L') on which d₋₎₂ is non-decreasing. We also prove that the respective intersections of M' and M'' with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set M L and describe two new maximal gaps of M nearby.