In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers.

中文翻译：

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广义马尔可夫数

在本文中，我们介绍了广义马尔可夫数，并将离散马尔可夫谱的经典马尔可夫理论扩展到广义马尔可夫数的情况。特别地，我们展示了这些数字的递归特性，并在马尔可夫谱中找到了相应的值。此外，我们构造了一个针对广义马尔可夫唯一性猜想的反例。建议的概括是基于数字的几何。它实质上使用了格子三角学和连续数的几何理论。