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 $f(x)\neq 0$ there exists a sequence $t_k\to\infty$ such that $\int_0^{t_k}f(T_sx)ds=0$ and $T_{t_k}x\in A$.