 |
|
Close Lattice Points on Circles |
Canad. J. Math. 61(2009), 1214-1238
Published:2009-12-01
Printed: Dec 2009
Printed: Dec 2009
- Javier Cilleruelo
- Andrew Granville
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 - Applications of sieve methods |