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Close Lattice Points on Circles

  Published:2009-12-01
 Printed: Dec 2009
  • Javier Cilleruelo
  • Andrew Granville
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Abstract

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 $tR^{1/3}$ on a circle of radius $R$, for any given $t>0$. In particular we prove that any arc of length $ (40 + \frac{40}3\sqrt{10} )^{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, $(\nu_{1,n},\nu_{2,n},\nu_{3,n},\nu_{4,n})$, each of which lies on an arc of length $ (40 + \frac{40}3\sqrt{10})^{\smash{1/3}}R_n^{\smash{1/3}}+o(1)$ on a circle of radius $R_n$.
MSC Classifications: 11N36 show english descriptions Applications of sieve methods 11N36 - Applications of sieve methods
 

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