2024/11/11 by Harold Erazo, Erazo, Harold, Davi Lima +7
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2411.06933
openalex publication_date 2024/11/11 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28
The Lagrange and Markov spectra L and M describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, L∩ (0,3) = M∩ (0,3) is a discrete set of explicit quadratic irrationals accumulating only at 3. In this article, we show that the statement above ceases to be true immediately after 3: in particular, L∩ (3,3+ε)≠ M∩ (3,3+ε) for all ε>0, and thus inf(M∖ L)=3. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of (M∖ L)∩ (3,3+ε) implying that \liminfε→ 0 (dimH((M∖ L)∩(3,3+ε)))/(dimH(M∩ (3,3+ε)))≥ (1)/(2) and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--Wüstholz theorem on linear forms in logarithms of algebraic numbers.