  We consider the set $\mathcal{R}_\mathrm{io}$ of points returning infinitely
many times to a sequence of shrinking targets around themselves. Under
additional assumptions we improve Boshernitzan's pioneering result on the speed
of recurrence. In the case of the doubling map as well as some linear maps on
the $d$ dimensional torus, we even obtain a dichotomy condition for
$\mathcal{R}_\mathrm{io}$ to have measure zero or one. Moreover, we study the
set of points eventually always returning and prove an analogue of
Boshernitzan's result in similar generality.
