Consider sequence $a_1=x$, $a_{n+1}=|a_n|^p\sin a_n$. Denote $S$ the set $\Big\{x\ \Big\vert \lim\limits_{n\to\infty}a_n=0\Big\}$, whats the value of $$\lim_{x\to\infty}\frac1{2x}\mu([-x,x]\cap S)?$$ (where $\mu(M)$ denotes the measure of $M$)
I tried starting with $p=2$. I plotted the graph of $\ln |a_n|$ with many $x$. It seems that almost every $x$ will make the limit tends to $0$.
For $0\le p< 1$, I can prove that the limit is $1$. All of the $x$ will make $\lim\limits_{n\to\infty}a_n$ tend to $0$.
Also, I can calculate that for $p=1$ the limit we want to find is also $1$.