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.
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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