Let $f(x,y)=ax^2+by^2+cxy$ be a real quadratic form $x,y\in \mathbb{R}$. Suppose that there is two sequence of natural numbers $(p_{1,n})$ and $(q_{2,n})$ such that $$ f(p_{1,n},q_{2,n})=O(p_{1,n}+q_{2,n})+o(n) $$
then if the sequence $(\frac{q_{2,n}}{p_{1,n}})$ has more than three accumulation points follows that $f$ is zero along the lines in $\mathbb{R}^2$ of slope this accumulations points (take the limit in the expression above divide by $p_{1,n}$ on the subsequence wich converges). So $f$ need to be zero because an equation of degree two in one variable cannot have more than three roots if is non-zero.
So my question is, I have tried do a generalization of this argument for a real quadratic form but instead of two variables $n-$variables but without success, it's possible show this fact with another argument or the same with some change?.