I am stuck on the following exercise:

Show that for any non-degenerate quadratic form $P$ over $\mathbb{R}$, that is either indefinite or positive definite, exists an integer point $(x,y) \in \mathbb{Z}^2 \setminus \{0\}$ such that

$$0 < P(x,y) < 2\sqrt{\lvert \det(P) \rvert}.$$

From the lecture I know about Minkowski's Theorem, but I do not see how to use this here. Could you please help me?

EDIT: The part with positive definite forms was already solved here:

Minkowski's convex body theorem and binary quadratic forms

So we are only left with the case of indefinite forms.



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