Abstract
In recent work, Miezaki introduced the notion of a spherical T-design in \(\mathbb {R}^2\), where T is a potentially infinite set. As an example, he offered the \(\mathbb {Z}^2\)-lattice points with fixed integer norm (a.k.a. shells). These shells are maximal spherical T-designs, where \(T=\mathbb {Z}^+\setminus 4\mathbb {Z}^+\). We generalize the notion of a spherical T-design to special ellipses, and extend Miezaki’s work to the norm form shells for rings of integers of imaginary quadratic fields with class number 1.
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Notes
We do not use the term ellipse to avoid possible confusion that might arise with the term elliptical.
References
Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, Graduate Texts in Mathematics, 137. Springer, New York (1992)
Bannai, E.: Spherical t-designs which are orbits of finite groups. J. Math. Soc. Jpn. 36(2), 341–354 (1984)
Bannai, E., Okuda, T., Tagami, M.: Spherical designs of harmonic index t. J. Approx. Theory 195, 1–18 (2015)
Chen, X., Frommer, A., Lang, B.: Computational existence proofs for spherical t-designs. Numer. Math. 117(2), 289–305 (2011)
Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata. 6(3), 363–388 (1977)
Hayashi, A., Hashimoto, T., Horibe, M.: Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A 73, 3 (2005)
Iwaniec H.: Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI, 1997. xii+259 pp
Miyake, T.: Modular forms, Translated from the 1976 Japanese original by Yoshitaka Maeda. Springer Monographs in Mathematics. Springer, Berlin (2006)
Miezaki, T.: On a generalization of spherical designs. Discrete Math. 313(4), 375–380 (2013)
Seki, G.: On some nonrigid spherical t-designs. Mem. Fac. Sci. Kyushu Univ. Ser. A 46(1), 169–178 (1992)
Acknowledgements
I would like to thank Prof Ken Ono for suggesting me this problem and guiding through. I also thank Will Craig and Wei-Lun Tsai for reviewing my paper and giving useful comments. I thank Matthew McCarthy for helping me with Sage Math. Lastly, I would like to thank the reviewers for their useful comments.
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Pandey, B.V. Modular forms and ellipsoidal T-designs. Ramanujan J 58, 1245–1257 (2022). https://doi.org/10.1007/s11139-022-00572-6
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DOI: https://doi.org/10.1007/s11139-022-00572-6