  We present shrinking targets results for general systems with the emphasis on
applications for IETs (interval exchange transformations) $(J,T)$, $J=[0,1)$.
In particular, we prove that if an IET $(J,T)$ is ergodic (relative to the
Lebesgue measure $\lam$), then the equality \[ \liminf_{n\to\infty}\limits n
|T^n(x)-y|=0 \tag{A1} \] holds for $\lam\ttimes\lam$-a. a. $(x,y)\in J^2$. The
ergodicity assumption is essential: the result does not extend to all minimal
IETs. The factor $n$ in (A1) is optimal (e. g., it cannot be replaced by $n
\ln(\ln(\ln n))$.
  On the other hand, for Lebesgue almost all 3-IETs $(J,T)$ we prove that for
all $\eps>0$ \[ \liminf_{n\to\infty}\limits n^\eps |T^n(x)-T^n(y)|=
\infty,\quad \text{for Lebesgue a. a.} (x,y)\in J^2. \tag{A2} \] This should be
contrasted with the equality $ \liminf_{n\to\infty}\limits |T^n(x)-T^n(y)|=0, $
for a. a. $(x,y)\in J^2$, which holds since $(J^2, T\times T)$ is ergodic
(because generic 3-IETs $(J,T)$ are weakly mixing).
  We also prove that no 3-IET is strongly topologically mixing.
