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Vector Addition

A circle K ce...

Question

A circle K centered at $(0, 0)$ is given. Prove that for every vector $(a_{1}, a_{2})$ there is a positive integer n such that the circle K translated by the vector $n (a_{1}, a_{2})$ contains a lattice point (i.e., a point both of whose coordinates are integers)

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Solution

Let r be the radius of K and $s > \sqrt{2} / r$ an integer. Consider the points $A_{k} (k a_{1} - [k a_{1}], k a_{2} - [k a_{2}])$ , where $k = 0, 1, 2, . . . ., s^{2}$ . Since all these points are in the unit square, two of them, say $A_{p}, A_{q}, q > p$ , are in a small square with side $1 / s$ , and consequently $A_{p} A_{q} \leq \sqrt{2} / s < r$ . Therefore, fir $n = q - p, m_{2} = [q a_{1}] - [p a_{q}]$ and $m_{2} = [q a_{2}] - [p a_{2}]$ the distance between the points $n (a_{1}, a_{2})$ and $(m_{1}, m_{2})$ is less than r, i.e., the point $(m_{1}, m_{2})$ is in the circle $K + n (a_{1}, a_{2})$ .

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