Close Lattice Points on Circles
Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1214-1238
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We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length $t{{R}^{1/3}}$ on a circle of radius $R$ , for any given $t\,>\,0$ . In particular we prove that any arc of length ${{\left( 40\,+\frac{40}{3}\sqrt{10} \right)}^{1/3}}\,{{R}^{1/3}}$ on a circle of radius $R$ , with $R\,>\,\sqrt{65}$ , contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points, $\left( {{v}_{1,n}},\,{{v}_{2,n}},\,{{v}_{3,n}},\,{{v}_{4,n}} \right)$ , each of which lies on an arc of length ${{\left( 40+\frac{40}{3}\sqrt{10} \right)}^{1/3}}R_{n}^{1/3}\,+\,o\left( 1 \right)$ on a circle of radius ${{R}_{n}}$ .
Cilleruelo, Javier; Granville, Andrew. Close Lattice Points on Circles. Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1214-1238. doi: 10.4153/CJM-2009-057-2
@article{10_4153_CJM_2009_057_2,
author = {Cilleruelo, Javier and Granville, Andrew},
title = {Close {Lattice} {Points} on {Circles}},
journal = {Canadian journal of mathematics},
pages = {1214--1238},
year = {2009},
volume = {61},
number = {6},
doi = {10.4153/CJM-2009-057-2},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2009-057-2/}
}
TY - JOUR AU - Cilleruelo, Javier AU - Granville, Andrew TI - Close Lattice Points on Circles JO - Canadian journal of mathematics PY - 2009 SP - 1214 EP - 1238 VL - 61 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2009-057-2/ DO - 10.4153/CJM-2009-057-2 ID - 10_4153_CJM_2009_057_2 ER -
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