Quantum numbers and markov polynomials

Quantisation (or q-deformation) of numbers goes back to Euler and Gauss, but in general it is far from being a well-defined procedure. In this thesis we discuss the quantisation of the real numbers proposed recently by Morier-Genoud and Ovsienko, who were inspired by combinatorics and cluster algebras. One of the open problems in this area is to prove the conjectural lower bound for the radius of convergence of the corresponding power series. We consider the opposite case of which rational numbers have the maximal radius of convergence, 1. By combining the theory of q-rational numbers together with classical results of Kronecker, Serret and Fatou we are able to shed more light on the classification of such rational numbers (Kronecker fractions). Meanwhile Markov polynomials, which are components of the solutions to a generalisation of the Markov equation, can also be considered as a quantisation of the Markov numbers, previously studied in the context of cluster algebras. We study the coefficients of these polynomials as a function on the corresponding Newton polygon. We propose a number of conjectures, which we were able to prove in several special cases. In particular, we found an analogue of Cohn matrices and relate them to the combinatorial interpretation of Markov polynomials.