Generalized discrete Lagrange–Markov spectra

Abstract: This talk concerns a discrete extension of the classical Lagrange and Markov spectra, motivated by generalized Markov equations. In the classical case, the discrete spectral values below $3$ are organized by Markov numbers and are described through continued fractions, Christoffel words, and Cohn matrices. I will explain how an analogous picture can be developed for generalized Markov numbers arising from $$ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy =(3+k_1+k_2+k_3)xyz. $$ For each generalized Markov number, one obtains an explicit spectral value which is realized both as the Lagrange constant of a quadratic irrational and as the Markov constant of an indefinite binary quadratic form with rational coefficients. The emphasis of the talk will be on the main idea of the construction: generalized Cohn matrices and symbolic sequences coming from straight-line codings play the role classically played by Christoffel words and Cohn matrices. The aim is to present a combinatorial and matrix-theoretic framework for viewing classical and generalized discrete Diophantine spectra in a unified way.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)