Abstract
For any given full rank lattice \(\Lambda \) and natural number K, we consider the point set \(P(K)=\Lambda /K\cap (0,1)^2\), with \(N=\#P(K)\approx K^2\), and bound the spherical cap discrepancy of the projection of these points under the Lambert map to the unit sphere. The bound is of order \(1/\sqrt{N}\), with leading coefficient given explicitly and depending on \(\Lambda \) only. The proof is established using a lemma that bounds the number of intersections of certain curves with fundamental domains that tile \(\mathbb {R}^2\), and even allows for local perturbations of \(\Lambda \) without affecting the bound, proving to be suitable for numerical applications. A special case yields the smallest constant for the leading term of the cap discrepancy for deterministic algorithms up to date.
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Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Code Availability
The code to produce the point distributions in Sect. 3 will be made available on the authors website: www.damirferizovic.wordpress.com.
Notes
Here \(0\le \phi <2\pi \) and \(0<\theta <\pi \).
See the Gauss circle problem, where Lemma 2.7 could be applied.
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Acknowledgements
The author thanks the anonymous referees for valuable suggestions and for providing the proof of Lemma 2.3. Further, thanks to Michelle Mastrianni for improving the text of the document, thanks to Dmitriy Bilyk and Arno Kuijlaars for skimming the paper and useful remarks; and thanks to the people involved with GNU Octave, Inkscape, LibreOffice, and TeXstudio, who made this document and many others possible.
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The work was supported by the Methusalem grant METH/21/03—long term structural funding of the Flemish Government.
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Ferizović, D. Spherical Cap Discrepancy of Perturbed Lattices Under the Lambert Projection. Discrete Comput Geom 71, 1352–1368 (2024). https://doi.org/10.1007/s00454-023-00547-4
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DOI: https://doi.org/10.1007/s00454-023-00547-4