Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve.
Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r \} $
Let $\gamma_{\mathbb{r,Z}}$ be the set $\gamma_{r} (\mathbb{R}) \cap \mathbb{Z}^{2}$.
Let $t >0$ and $\gamma_{r,\mathbb{Z}}^{t}$ be the set $\gamma_{r}([-t,t]) \cap \mathbb{Z}^{2}$.
Let $\omega : \mathbb{Z}^{2} \to \{ 0 , 1\}$ be a function.
Definition : $\omega$ is algebraically normal if for all algebraic curve $\gamma$ and $\forall r \geq 0$ such that $\vert \gamma_{\mathbb{r,Z}} \vert = \infty$ : $$ \lim\limits_{t \to \infty} \frac{\vert \omega^{-1}(0) \cap \gamma_{\mathbb{r,Z}}^{t} \vert}{\vert \gamma^{t}_{\mathbb{r,Z}} \vert} = \frac{1}{2} $$
Is there exist such an algebraically normal function $\omega$ ?
Intuitively, it seems there are uncountably many such $\omega$, but in practice, can we prove the existence ?