Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals

We recall theorems by Krygin, Atkinson, Shneiberg and propose the following assertion. Let $T_t$ be an ergodic flow on $(X,\mu )$, let a function $f$ on $X$ have zero mean, and $\mu (A)>0$ for $A\subset X$. Then for almost all $x\in A$ with $f(x)\mathrm{\ne }0$ there exists a sequence $t_k\to \mathrm{\infty }$ such that ${\int }_0^{t_k}f(T_{s}x)ds=0$ and $T_{t_k}x\in A$.