The intersection condition for \(\langle s\rangle\)-density systems of paths (Q930103)

The existence of systems \(\{E_x\subset\mathbb R:x\in\mathbb R\}\) of sets strongly porous at \(x\) such that \(x\) is an \(\langle s\rangle\)-density point of \(E_x\) for every \(x\in\mathbb R\) (i.e., \(\lim_{n\to\infty}\frac{| E_x\cap [x-1/{s_n},x+1/{s_n}]| }{2/{s_n}}=1\), where \(s=(s_n)\)) and \(\{E_x\subset\mathbb R:x\in\mathbb R\}\) does not have one of two relatively weak intersection properties is shown if \(\liminf_{n\to\infty}\frac{s_n}{s_{n+1}}=0\).