, 2010

Let $L$ be a line in the plane, and let $\pi_1$ and $\pi_2$ be the corresponding open half-planes. We set
$$f_L(P) = \begin{cases} 1 & \text{if } P \in \pi_1 \cup L, \\ -1 & \text{if } P \in \pi_2. \end{cases}$$
We will show that $f_L$ is a perfect function. Then the family $\{f_L : (0,0) \in L\} \subset \mathcal{F}$ consists of infinitely many perfect functions that are not translates of each other.

Let $g$ be a function in $\mathcal{F}$ differing from $f_L$ at finitely many points, and let
$$K_{\pm} = \{(P, Q) : f_L(P)f_L(Q) - g(P)g(Q) = \pm 2 \text{ and } 0 < d(P, Q) < 2010\}.$$
Then
$$\sum_{0 < d(P,Q) < 2010} \frac{f_L(P)f_L(Q) - g(P)g(Q)}{d(P,Q)} = \sum_{(P,Q) \in K_+} \frac{2}{d(P,Q)} - \sum_{(P,Q) \in K_-} \frac{2}{d(P,Q)}$$

We will prove that this expression is nonnegative by defining an injection $m: K_- \to K_+$ that satisfies $d(m(P, Q)) = d(P, Q)$.

Let $(P, Q) \in K_-$. Then $f_L(P) \neq f_L(Q)$ and $g(P) = g(Q)$. Consider the line $\ell$ passing through the points $P$ and $Q$. Let $P_0 = P$, $P_1 = Q$, and let $P_i$ be the unique point on $\ell$ such that $d(P_i, P_{i-1}) = d(P, Q)$ and $P_i \neq P_{i-2}$ for $i \ge 2$, and $d(P_i, P_{i+1}) = d(P, Q)$ and $P_i \neq P_{i+2}$ for $i \le -1$.

Since $f_L$ and $g$ differ at finitely many points, we know that $f_L(P_i) = g(P_i)$ for all $i$ with sufficiently large $|i|$. In particular, $g$ changes sign finitely many times, but at least once on $\ell$. Let $k$ be the smallest integer such that $g(P_k) \neq g(P_{k+1})$. We define $m(P, Q) = (P_k, P_{k+1})$.

$(P_k, P_{k+1}) \in K_+$ as $f_L(P_k)f_L(P_{k+1}) - g(P_k)g(P_{k+1}) = 1 - (-1) = 2$, and we have $d(P_k, P_{k+1}) = d(P, Q)$ by construction. Finally, since $(P, Q)$ is the only pair among $(P_i, P_{i+1})$, $i \in \mathbb{Z}$, satisfying $f_L(P_i) \neq f_L(P_{i+1})$, $m$ is injective.