Asymptotic number of lattice points in an annulus

Question

Let $\Gamma$ be a lattice in $\mathbb{R}^2$ and for $n\in \mathbb{N}$, let $K(n) = |\lbrace \gamma \in \Gamma: |\gamma| < n\rbrace|$.

How do I show $K(n) - K(n-1)$ is $O(n)$?

By the theory of integration I know that $\lim\limits_{n \to \infty}\frac{K(n)}{\pi n^2} = 1$. But this gives nothing related to what I want. Please help me proceed.

Answer 1

A sketch.

A circle of radius $n-3$ has area $\pi(n-3)^2$. So at least this many unit square touches it. Each square touches it must be entirely within the circle of radius $n-1$. This give a lower bound.

Similarly a circle of radius $n$ contains at most $\pi (n+2)^2$ squares.

So difference is at most $O(n)$ squares.