Reference : Lemma 5.3 from "Groups acting on circle" by Étienne Ghys.
Let $f:\mathbb{Z}\rightarrow \mathbb{R}$ be a quasi-homomorphism, i.e $|f(a+b)-f(a)-f(b)|\leq D$, $\forall$ $a$ and $b$ in $\mathbb{Z}$ ($\mathbb{R}$ and $\mathbb{Z}$ are here considered as additive groups and so you see the plus sign). I have to prove that there exists a unique number $\tau\in\mathbb{R}$ such that $f(n)-n\tau$ is bounded.
I have separately shown the uniqueness part and have found bounds that work in following restrictive cases :
- (i) If $n$ is positive, I have bounded it above by $f(0)$ using $\tau=f(1)+D$.
- (ii) If $n$ is positive, I have bounded it below by $f(0)$ using $\tau=f(1)-D$.
- (iii) If $n$ is negative, I have bounded it above by $f(0)$ using $\tau=f(-1)+D$.
- (iv) If $n$ is negative, I have bounded it below by $f(0)$ using $\tau=f(-1)-D$.
I understand that I have always used $f(0)$ to bound them, but that is only what I can see since I am splitting $n$ as $n$ times the generator $1\in\mathbb{Z}$. Please help me bound this universally using just one unique $\tau$.