Generalised Markov numbers

TL;DR

This paper introduces generalized Markov numbers, extending classical Markov theory to a broader context using geometric methods, and provides new recursive properties, spectrum values, and a counterexample to a conjecture.

Contribution

It presents a novel generalization of Markov numbers based on geometry of numbers, including recursive properties and a counterexample to the uniqueness conjecture.

Findings

Established recursive properties for generalized Markov numbers

Identified specific values in the Markov spectrum for the generalization

Constructed a counterexample to the generalized Markov uniqueness conjecture

Abstract

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.