Geodesic
Close Lattice Points on Circles
Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1214-1238

Voir la notice de l'article provenant de la source Cambridge University Press

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 ${{R}_{n}}$ .
DOI : 10.4153/CJM-2009-057-2
Mots-clés : 11N36 
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
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[1] [1] Bombieri, E. and Pila, J., The number of integral points on arcs and ovals. Duke Math. J. 59(1989), 337–357. Google Scholar

[2] [2] Cilleruelo, J., Arcs containing no three lattice points. Acta Arithmetica 59(1991), 87–90. Google Scholar

[3] [3] Cilleruelo, J. and Córdoba, A., Trigonometric polynomials and lattice points. Proc. Amer. Math. Soc. 115(1992), 899–905. Google Scholar

[4] [4] Cilleruelo, J. and Granville, A., Lattice points on circles, squares in arithmetic progressions and sumset of squares. In: Additive Combinatorics, CR MProc. Lecture Notes 43, American Mathematical Society, Providence, RI, 2007, pp. 241–262. Google Scholar

[5] [5] Cilleruelo, J. and Granville, A., Close divisors. In preparation. www.dms.umontreal.ca/∼andrew/preprints.html Google Scholar

[6] [6] Cilleruelo, J. and Tenenbaum, G., An overlapping theorem with applications. Publicacions Mathematiques, 51(2007), 107–118. Google Scholar

[7] [7] Elkies, N. D., Rational points near curves and small nonzero |x 3 − y2| via lattice reduction. In: Algorithmic Number Theory, Lecture Notes in Comput. Sci. 1838 Springer-Verlag, New York 2000, pp. 33–63. Google Scholar

[8] [8] Ellenberg, J. and Venkatesh, A., On uniform bounds for rational points on nonrational curves. Int. Math. Res. Not. 35(2005), 2163–2181. Google Scholar

[9] [9] Heath-Brown, R., The density of rational points on curves and surfaces. Ann. of Math. 155(2002), 553–595. Google Scholar

[10] [10] Zygmund, A., A Cantor-Lebesgue theorem for double trigonometric series. Studia Matematica 43(1972), 173–178. Google Scholar

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