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Lattice points in d-dimensional spherical segments

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Abstract

We study lattice points in d-dimensional spheres, and count their number in thin spherical segments. We found an upper bound depending only on the radius of the sphere and opening angle of the segment. To obtain this bound we slice the segment by hyperplanes of rational direction, and then cover an arbitrary segment with one having rational direction. Diophantine approximation can be used to obtain the best rational direction possible.

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Notes

  1. Here and in what follows \(f(m)\ll g(m)\) means that f(m)/g(m) is bounded as m tends to infinity.

  2. Here and elsewhere f(R)=o(g(R)) means that f(R)/g(R) tends to 0 as R tends to infinity.

  3. In fact, Linnik aslo proved the statement replacing the constraint mod 5 by a similar one mod p, for any fixed prime p.

  4. To the best of my knowledge, an upper bound for \(\kappa _d\) when \(d\ge 4\) has not been explicitly studied in the literature, but a bound like \(\kappa _d(R) \ll R^{d-3+\epsilon }\) would be expected to be amenable to the circle method for \(d\ge 5\), since the intersection of a hyperplane and a \((d-1)\)-sphere is a \((d-2)\)-sphere.

  5. Here \(\ll _\beta \) means that the implied constant depends on \(\beta \).

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Acknowledgements

Thanks to Riccardo Maffucci for being my supervisor in this project, who has been indispensable in every sense. Thanks also to the LMS and Oxford University Math Institute for providing funding for this project. I would also like to thank Zeév Rudnick and R. Heath-Brown for generously answering my questions about their work.

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  1. University of Oxford, Oxford, Oxfordshire, UK

    Martin Ortiz Ramirez

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  1. Martin Ortiz Ramirez
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Correspondence to Martin Ortiz Ramirez.

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Communicated by Adrian Constantin.

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Ortiz Ramirez, M. Lattice points in d-dimensional spherical segments. Monatsh Math 194, 167–179 (2021). https://doi.org/10.1007/s00605-020-01447-y

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