Let G be a domain in $\mathbb{R}^3$ with the following properties:

- bounded
- finitely connected
- positive Gaussian curvature
- 2-times continuously differentiable boundary $\partial G$

Let $T>0$ and $G_T=\{Tx : x \in G \}$. We denote with $G^d_T$ the points with difference of less than $d>0$ from $G_T$ and write $G^d$ for the points with difference of less than $d>0$ from $G$.

I would like to compute the sum $$ \lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in (\mathbb{Z}^3\cap G_T^d\setminus G_T)}f(k), $$ where $f(k)$ is a function from $\mathbb{R}^3 $ to $\mathbb{R}$. In other words: I want to sum over every function value of $ f(k)$ for which $k\in \mathbb{Z}^3$ lies within the shell $G_T^d\setminus G_T$.

Some numerical experiments suggest that the number of points within the shell $G_T^d\setminus G_T$ depend on the thickness of the shell, the scaling factor T and the n-1 dimensional Hausdorff measure $\mu(\partial G)$ of $G$. More precisely $$ card \{k\in \mathbb{Z}^3: k\in G_T^d\setminus G^d \} \approx T^2 d \mu(\partial G) $$

It is clear that $$ \lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in (\mathbb{Z}^3\cap G_T^d\setminus G_T)} f(k)=\lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in \big(\frac{\mathbb{Z}^3}{T}\cap G^d\setminus G\big)} f(Tk). $$

My goal is to show that $$\lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in \big(\frac{\mathbb{Z}^3}{T}\cap G^d\setminus G\big)} f(Tk) = \int_{G^d \setminus G}\lim_{T\rightarrow \infty}f(T\omega)d\omega . $$ Therefore i need to proof that the number of points inside the shell is indeed of order $T^2 d \mu(G)$ and more importantly that the points are equidistributed inside the region $G^d\setminus G^d$. My problem is that I'm not sure how to prove those two statements.