Counting lattice points inside circle

Question

Let $p_1,p_2 \in \mathbb C$ be linearly independent over $\mathbb R$. Let $A$ denote the area of the set $\{t_1p_1+t_2p_2 : t_1,t_2 \in [0,1]\}$ in the complex plane. Let $f(R):=| \{z \in \mathbb Zp_1+\mathbb Zp_2 : |z| \le R\}|$ . Then how to show that $f(R)=\pi R^2/A+O(R)$ as $R\to \infty $ ?

If $p_1=1, p_2=i$, then I can prove the desired estimate. From that can we derive the desired estimate for any $\mathbb R$-linearly independent $p_1,p_2 \in \mathbb C$ ?

Please help.

Answer 1

Let $l = \frac 12\max\{|p_1|, |p_2|, |p_1 + p_2|\}$, which is a fixed constant given the lattice.

Denote $I_R = \{z \ |\ z \in \mathbb Zp_1 + \mathbb Z p_2, |z| \leq R\}$.

Denote $F = \{t_1p_1 + t_2p_2 \ | \ t_1, t_2 \in [-\frac12, \frac12]\}$

Also denote $\overline P_R = \bigcup_{z \in I_R} (z+F)$.

Let $D_R$ be the closed disk centered at origin with radius $R$.

An illustration is shown below.

Claim: For $R > l$, $D_{R-l} \subseteq \overline P_R \subseteq D_{R+l}$.

Your question then follows from the fact that the area of $I_R = f(R)A$.