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Convexity and Aigner's Conjectures
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Convexity and Aigner's Conjectures
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Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic $$x^2 + y^2 + z^2 - 3x y z = 0.$$ A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. One can associate to each a positive rational number a Markov number in a natural way. We give a new unified proof of certain conjectures from Martin Aigner's book, Markov's Theorem and 100 Years of the Uniqueness Conjecture. Our proof relies on a relationship between Markov numbers and the lengths of closed simple geodesics on the punctured torus discovered by H. Cohn.
Forward citations
Cited by 5 Pith papers
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McShane-Rivin norm balls and simple-length multiplicities
Simple closed geodesics of exactly length L on a once-punctured torus number at most O((log L)^2), improving the classical O(L^{2/3}) boundary-lattice bound and Markoff-fiber bounds to O((log log m)^2).
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Generalized discrete Markov spectra
The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a ...
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McShane-Rivin norm balls and simple-length multiplicities
On hyperbolic once-punctured tori the multiplicity of any simple geodesic length L is at most C (log L)^2, improving prior logarithmic bounds for Markoff numbers.
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Orderings of Generalized k-Markov Numbers
Generalized k-Markov numbers grow monotonically along more random lines as k increases, supporting a k-analog of Frobenius' uniqueness conjecture.
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Orderings of k-Markov Numbers
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.
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