Generalised Markov numbers

Karpenkov, Oleg ORCID: 0000-0002-3358-6998 and van-Son, Matty ORCID: 0000-0001-8576-6758 (2020) Generalised Markov numbers. Journal of Number Theory, 213. pp. 16-66.

Text 1809.01688v1.pdf - Submitted version Download (463kB)

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.

Item Type:	Article
Additional Information:	50 pages
Uncontrolled Keywords:	Geometry of continued fractions, Perron identity, Binary quadratic indefinite form, Markov spectrum
Depositing User:	Symplectic Admin
Date Deposited:	07 Sep 2018 10:00
Last Modified:	19 Jan 2023 01:25
DOI:	10.1016/j.jnt.2020.01.010
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3025928