It’s a problem from Viana’s book "Foundations of Ergodic Theory"
Show that if $f:[0,1]\to [0,1]$ is a measurable transformation preserving the Lebesgue measure $m$ then $m-$almost every point $x\in [0,1]$ satisfies $$\liminf_{n} ~n\lvert f^n(x)-x\rvert \leq 1$$
The author gave us some hint (but unfortunately, I still cannot complete it):
Suppose that the conclusion is not true. Then $\exists k\geq 1$ and $b>1$ such that the set $$B=\left\{x\in[0,1]:n\lvert f^n(x)-x\rvert>b ~\textit{ for every }~n\geq k\right\}$$ has positive measure. Let $a\in B$ be a density point of $B$. Consider $E=B\cap B(a,r)$, for $r$ small. Get a lower estimate for the return time to $E$ of any point of $x\in E$ and use the Kac theorem to reach a contradiction.
Kac theorem tells us : $$\int_E \rho_E d\mu=\mu([0,1])-\mu(E_0^*)$$ where $\rho_E(x)=\min\left\{ n\geq 1:f^n(x)\in E\right\}$, $E_0^*=\left\{x\in [0,1]:f^n(x)\notin E,~\forall n\geq 1\right\}$.
I don’t know how to estimate the lower bound of $\rho_E$, and don’t know what the role of the density point $a$. And I thought this problem for a long time. I do hope someone can give me suggestions. Thank you!