Suppose $D\subset\Bbb R$ such that $D\cap P\neq\emptyset$ for each nonempty perfect set $P\subset\Bbb R.$ Notice that $D$ need not to be a Bernstein set. Clearly, $D$ intersects each perfect set in continuum many points.
Is $r D\cap P\neq\emptyset$ for every nonempty perfect set $P$ and $0\neq r\in\Bbb R$? $rD$ means rescaling $D$ by $r$.
I know rescaling the perfect set is still the perfect set. Any idea?