广义离散马尔可夫谱
Generalized discrete Markov spectra
AI总结：
本文提出广义马尔可夫理论，研究广义马尔可夫-拉格朗日谱，通过广义马尔可夫数扩展经典离散马尔可夫谱，利用蛇图替代Christoffel-word形式，构建广义马尔可夫方程谱，并证明其结构特性与贡献。
摘要：
我们发展了广义马尔可夫理论，用于马尔可夫-拉格朗日谱和马尔可夫谱。经典的离散马尔可夫谱由马尔可夫数支配，即正整数解于马尔可夫方程中出现。我们证明这种关系允许一个簇-组合扩展，由广义马尔可夫数支配。通过将Christoffel-word形式替换为蛇图，我们构造了与广义马尔可夫方程 \[ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz \] 相关的广义离散马尔可夫谱。这些谱中的每个元素同时作为二次无理数的拉格朗日常数和实不定二元二次型的马尔可夫常数。我们还证明了这些谱的结构结果，确定了它们在弗里曼常数以下的过渡区间内的贡献，并识别了从具有无理斜率的正则线获得的边界值，再次将其作为拉格朗日常数和马尔可夫常数实现。
英文摘要：
We develop a generalized Markov theory for the Markov--Lagrange and Markov spectra. The classical discrete Markov spectrum is governed by Markov numbers, the positive integers occurring in solutions of the Markov equation. We show that this relation admits a cluster-combinatorial extension governed by generalized Markov numbers. Replacing the Christoffel-word formalism by snake graphs, we construct generalized discrete Markov spectra attached to the generalized Markov equations \[ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz. \] Every element of these spectra is realized simultaneously as a Lagrange constant of a quadratic irrational and as a Markov constant of a real indefinite binary quadratic form. We also prove structural results for these spectra, determine their contribution in the transition interval below Freiman's constant, and identify the boundary value obtained from regular lines of irrational slope, again realizing it both as a Lagrange constant and as a Markov constant.
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