Strong approximation and Diophantine properties of Markoff triples

Transitivity properties of the group of morphisms generated by Vieta involutions on the solutions to the Markoff equation modulo primes are established, yielding forms of strong approximation for the Markoff surface. These are applied to show that almost all Markoff numbers are highly divisible.

Authors

Alexander Gamburd
Jean Bourgain
Peter Sarnak

Institutions

The Graduate Center, CUNY (US)
Princeton University (US)
Institute for Advanced Study (US)

Publication Details

Journal: Journal of the American Mathematical Society
Published: 2025-08-29
DOI: https://doi.org/10.1090/jams/1061
Primary Topic: Mathematical Dynamics and Fractals
Type: article
Field-Weighted Citation Impact: 0.00

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article

Strong approximation and Diophantine properties of Markoff triples

Alexander Gamburd, Jean Bourgain, Peter Sarnak

Aug 29, 2025

Journal of the American Mathematical Society

Mathematical Dynamics and Fractals

article

Strong approximation and Diophantine properties of Markoff triples

Alexander Gamburd, Jean Bourgain, Peter Sarnak

August 29, 2025article en

Abstract

Journal of the American Mathematical SocietyVol. 39(1)

The Graduate Center, CUNY (US), Princeton University (US), Institute for Advanced Study (US)

Openalex Percentile: Top 71%

Mathematical Dynamics and Fractals

DOI Source OpenAlex

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