Markov spectra for modular billiards


The connection between Markov’s theory of minima of indefinite binary quadratic forms and hyperbolic geodesics is well-known. We introduce some new analogues of the Markov spectrum defined in terms of modular billiards and consider the problem of characterizing that part of the spectrum below the lowest limit point.

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The second author thanks Alex Kontorovich for some enlightening discussions on the topics of this paper. The authors thank the referee for several constructive comments that have improved the exposition of this paper.

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Correspondence to Nickolas Andersen.

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In memory of Harvey Cohn (1923–2014)

Supported by NSF Grant DMS 1701638.

Communicated by Kannan Soundararajan.

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Andersen, N., Duke, W. Markov spectra for modular billiards. Math. Ann. 373, 1151–1175 (2019).

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