Let $\rho: \mathbb{Z} \rightarrow \mathbb{Z}$ is a map with the property: $E^{+}_{\rho} = \{\rho(m + n) - \rho(m) - \rho(n) \in \mathbb{Z} \mid m,n \in \mathbb{N}\}$ is a finite set and $\rho(-n) = -\rho(n)$ for all $n \in \mathbb{Z}$. We call such maps are odd slopes.
- My promplem:
If $\rho$ is an odd slope and there is an infinite set $X \subset \mathbb{N}$ such that $\rho(X)$ is a finite set, then $\rho(\mathbb{N})$ is a finite set.
- My attempt:
From $\rho(X)$ is finite, $\{\rho(m + n) \in \mathbb{Z} \mid m,n \in X\}$ is finite by $E^{+}_{\rho}$. $A = \{k_m \in \mathbb{N} \mid m \in \mathbb{N}\}$ is finite where
$k_m := \min \{k \in \mathbb{N} \mid \exists n \in X: m + k = n\}$
$A$ is finite then $\rho(A)$ is finite therefore, for all $m \in \mathbb{N}$, exists $k_m, n \in \mathbb{N}$ such that: exists $u \in E^{+}_{\rho}$, $v \in \rho(X)$ and $w \in \rho(A)$:
$\rho(m) = \rho(m + k_m) - \rho(k_m) - u = \rho(n) - \rho(k_m) - u = v - w - u$
takes its values in a finite set. Hence, the set $\{\rho(m) \in \mathbb{Z} \mid m \in \mathbb{N}\}$ is finite $\Leftrightarrow \rho(\mathbb{N})$ is finite.
- My question:
I can not prove $A$ is finite. Maybe my approach or my problem statement was wrong. I hope your help.
