## 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

- Ryuji Abe, Iain R. Aitchison, and Benoît Rittaud,
*Two-color Markoff graph and minimal forms*, Int. J. Number Theory**12**(2016), no. 4, 1093–1122. MR**3484300**, DOI 10.1142/S1793042116500676 - Ryuji Abe and Benoît Rittaud,
*On palindromes with three or four letters associated to the Markoff spectrum*, Discrete Math.**340**(2017), no. 12, 3032–3043. MR**3698093**, DOI 10.1016/j.disc.2017.07.010 - Byungchul Cha, Heather Chapman, Brittany Gelb, and Chooka Weiss,
*Lagrange spectrum of a circle over the Eisensteinian field*, Monatsh. Math.**197**(2022), no. 1, 1–55. MR**4368629**, DOI 10.1007/s00605-021-01649-y - Byungchul Cha and Dong Han Kim,
*Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum*, 2021, To appear in*Ann. Institut Fourier.*arXiv:1903.02882. - Thomas W. Cusick and Mary E. Flahive,
*The Markoff and Lagrange spectra*, Mathematical Surveys and Monographs, vol. 30, American Mathematical Society, Providence, RI, 1989. MR**1010419**, DOI 10.1090/surv/030 - Harold Erazo, Carlos Gustavo Moreira, Rodolfo Gutiérrez-Romo, and Sergio Romaña,
*Fractal dimensions of the Markov and Lagrange spectra near $3$*, 2022, arXiv:2208.14830. - G. A. Freĭman,
*Non-coincidence of the spectra of Markov and of Lagrange*, Mat. Zametki**3**(1968), 195–200 (Russian). MR**227110** - G. A. Freĭman,
*The initial point of Hall’s ray*, Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), Kalinin. Gosudarstv. Univ., Moscow, 1973, pp. 87–120, 121–125 (Russian, with English summary). MR**429771** - G. A. Freĭman,
*Diofantovy priblizheniya i geometriya chisel (zadacha Markova)*, Kalinin. Gosudarstv. Univ., Kalinin, 1975 (Russian). MR**485714** - Benedict H. Gross and Mark W. Lucianovic,
*On cubic rings and quaternion rings*, J. Number Theory**129**(2009), no. 6, 1468–1478. MR**2521487**, DOI 10.1016/j.jnt.2008.06.003 - Dong Han Kim and Deokwon Sim,
*The Markoff and Lagrange spectra on the Hecke group $\mathbf {H}_4$*, 2022, arXiv:206.05441. - Hans Günther Kopetzky,
*Über das Approximationsspektrum des Einheitskreises*, Monatsh. Math.**100**(1985), no. 3, 211–213 (German, with English summary). MR**812612**, DOI 10.1007/BF01299268 - A. V. Malyšev,
*Markov and Lagrange spectra (a survey of the literature)*, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**67**(1977), 5–38, 225 (Russian). Studies in number theory (LOMI), 4. MR**441876** - A. V. Malyšev,
*Markov and Lagrange spectra (survey of the literature)*, J. Soviet Math.**16**(1981), 767–788. - A. Markoff,
*Sur les formes quadratiques binaires indéfinies.*, Math. Ann.**15**(1879), 381–406. - A. Markoff,
*Sur les formes quadratiques binaires indéfinies*, Math. Ann.**17**(1880), no. 3, 379–399 (French). MR**1510073**, DOI 10.1007/BF01446234 - Asmus L. Schmidt,
*Minimum of quadratic forms with respect to Fuchsian groups. I*, J. Reine Angew. Math.**286(287)**(1976), 341–368. MR**457358**, DOI 10.1515/crll.1976.286-287.341 - Asmus L. Schmidt,
*Minimum of quadratic forms with respect to Fuchsian groups. II*, J. Reine Angew. Math.**292**(1977), 109–114. MR**457359**, DOI 10.1515/crll.1977.292.109 - L. Ya. Vulakh,
*The Markov spectra for Fuchsian groups*, Trans. Amer. Math. Soc.**352**(2000), no. 9, 4067–4094. MR**1650046**, DOI 10.1090/S0002-9947-00-02455-7

## 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