2022/09/26 by Luke Jeffreys, Jeffreys, Luke, Carlos Matheus +3 · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2209.12876
openalex publication_date 2022/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let L and M denote the Lagrange and Markov spectra, respectively. It is known that L⊂ M and that M∖ L≠\varnothing. In this work, we exhibit new gaps of L and M using two methods. First, we derive such gaps by describing a new portion of M∖ L near to 3.938: this region (together with three other candidates) was found by investigating the pictures of L recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of M∖ L and we improve upon a lower bound on the Hausdorff dimension of M∖ L obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of 0.593 on the dimension of M∖ L). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of M accumulating to Freiman's gap preceding the so-called Hall's ray [4.52782956616...,∞)⊂ L.