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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Diophantine approximation on conics
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by Evan M. O’Dorney
Proc. Amer. Math. Soc. 151 (2023), 1889-1905
DOI: https://doi.org/10.1090/proc/16273
Published electronically: February 17, 2023

Abstract:

Given a conic $\mathcal {C}$ over $\mathbb {Q}$, it is natural to ask what real points on $\mathcal {C}$ are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz’s theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha–Kim [Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum, arXiv: 1903.02882, 2021] and Cha–Chapman–Gelb–Weiss [Monatsh. Math. 197 (2022), pp. 1–55] shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general $\mathcal {C}$ and prove that their behavior is exhausted by the special family of conics $\mathcal {C}_n : XZ = nY^2$, which has symmetry by the modular group $\Gamma _0(n)$ and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh [Trans. Amer. Math. Soc. 352 (2000), pp. 4067–4094]. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of $\operatorname {Mat}^{2\times 2}(\mathbb {Z})$ and classifying invariant lattices in its $2$-dimensional representation.
References
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Bibliographic Information
  • Evan M. O’Dorney
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46617
  • MR Author ID: 1043604
  • ORCID: 0000-0002-7958-2060
  • Email: eodorney@math.nd.edu
  • Received by editor(s): May 27, 2022
  • Received by editor(s) in revised form: September 12, 2022
  • Published electronically: February 17, 2023
  • Communicated by: David Savitt
  • © Copyright 2023 by Evan M. O’Dorney
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 1889-1905
  • MSC (2020): Primary 11J06
  • DOI: https://doi.org/10.1090/proc/16273
  • MathSciNet review: 4556186