Lattice points in d-dimensional spherical segments
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Lattice points in d-dimensional spherical segments Martin Ortiz Ramirez1 Received: 7 November 2019 / Accepted: 16 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
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. Keywords Discrete geometry · Lattice points on spheres · Diophantine approximation Mathematics Subject Classification 11P21 · 11K60
1 Introduction The topic of lattice points in d-spheres has been extensively studied for centuries, starting with the two dimensional case of Gauss’s circle problem, which asks for the number of lattice points N (r ) inside a circle of radius r , with r 2 = n an integer. Gauss already knew that by geometric considerations N (r ) is close to the area of the circle πr 2 , and the problem transitioned to the study of the asymptotic behaviour of E(r ) = |N (r ) − πr 2 |. One natural question asks for the the infimum μ of the α such that E(r ) = O(r α ). Gauss proved that μ ≤ 1 by considering the length of the circumference, Sierpinski improved it to μ ≤ 2/3, and Van der Corput showed that μ < 2/3 [8, pp. 20–22]. They also conjectured that μ = 1/2. The best upper bound to date is μ ≤ 517/824 by Bourgain and Watt [4]. As for a lower bound, Hardy and Landau independently proved that μ ≥ 1/2.
Communicated by Adrian Constantin.
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Martin Ortiz Ramirez [email protected] University of Oxford, Oxford, Oxfordshire, UK
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M. Ortiz Ramirez
Gauss’s circle problem can be interpreted by considering the sum of all the r2 (m) for m ≤ n, where r2 (m) is the number of integral solutions to x 2 + y 2 = m. Unlike Gauss’s problem, where obtaining the correct order of magnitude is easy, it is more difficult for r2 (m), since it vanishes for arbitrarily large m. Relating r2 (m) to the number of divisors of m [9, Ch 16.10] yields r2 (m) m for any > 0.1 Consider the number of lattice points on a short circular arc. Cilleruelo and √ Cordoba [5] proved that on a circle of radius R, an arc of length no greater than 2R 1/2−1/(4l/2+2) contains at most l lattice points. Cilleruelo and Granville [6] proved that on an arc of length less √ 1/3 10) R 1/3 there are at most 3 lattice points, an improvement on a than (40 + 40 3 result by Jarnik [11], that in an arc of length less than or equal to than R 1/3 there are at most two lattice points. Moving to 3 dimensions, S(R) denoting the number of lattice points inside a sphere of radius R, we have the analogue of Gauss’s circle problem. By the same geometric considerations S(R) is of the order of 43 π R 3 , the volume of the sphere. Hence we are interested in E(R) = | 43 π R 3 − S(R)|. Szego [17] proved that E(R) = o(R(log R)1/
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