# Convergence of series defined on half space

Let E be an arbitrary ellipse in $$\mathbb{R}^2$$ with boundary $$\partial G$$. We denote by $$E_T$$ the dilated ellipse $$\{ Tx:x \in E \}.$$ Furthermore, let $$\tau(x)$$ be the unit outward normal vector of $$E$$ at $$x\in \partial G$$.

We assume that the sum $$\sum_{k\in \mathbb{Z}^2 : k\cdot \tau(x)<0}f(k)$$ is equal to A. [$$\cdot$$ is the standard scalar product]

I would like to proof that $$\lim_{T \rightarrow \infty} \sum_{ k\in \mathbb{Z}^2 : k\in E_T-Tx} f(k)=A.$$

Can anyone help me find an approach or give a reference to literature for these kinds of problems.

By definition, requiring that the sum : $$\sum_{k\in \mathbb{Z}^2 : k\cdot \tau(x)<0}f(k)$$ converges to $$A$$ means that for any non-decreasing sequence of finite subsets $$X_n \subset \{k\in \mathbb{Z}^2 : k\cdot \tau(x)<0\}$$, we have : $$\lim_{n\to +\infty} \sum_{k\in X_n} f(k) = A$$
Therefore, we just have to check that $$\{k\in\mathbb Z^2 |k\in E_T - Tx\}$$ is a finite subset of $$\{ k\in\mathbb Z^2 |k\cdot \tau(x) < 0\}$$ and that : $$\bigcup_{T\to \infty} \{k\in\mathbb Z^2 |k\in E_T - Tx\} = \{ k\in\mathbb Z^2 |k\cdot \tau(x) < 0\}$$