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

2024/05/31 by Harold Erazo, Erazo, Harold, Luke Jeffreys +3 · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Functional Equations Stability Results

paper · doi:10.48550/arxiv.2405.20581

Abstract

Let L and M denote the classical Lagrange and Markov spectra, respectively. It is known that L⊂ M and that M∖ L≠\varnothing. 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_loc(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_loc 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.

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• New examples of M∖ L: intruder sets

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