  In relation to the Erd\H os similarity problem (show that for any infinite
set $A$ of real numbers there exists a set of positive Lebesgue measure which
contains no affine copy of $A$) we give some new examples of infinite sets
which are not universal in measure, i.e. they satisfy the above conjecture.
These are symmetric Cantor sets $C$ which can be quite thin: the length of the
$n$-th generation intervals defining the Cantor set is decreasing almost doubly
exponentially. Further, we achieve to construct a set, not just of positive
measure, but of \textit{full measure} not containing any affine copy of $C$.
Our method is probabilistic.
