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The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

Springborn, Boris

Markov’s theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov’s theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the problems: How far can a straight line crossing a triangle stay away from the vertices? How far can it stay away from all vertices of the tessellation generated by this triangle? Definite binary quadratic forms are briefly discussed in the last section.
Published in: L' enseignement mathématique, 10.4171/LEM/63-3/4-5, European Mathematical Society
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  • This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.