A note on the O'Malley density property (Q1332600)

1. Let \(c\in (0,1]\) and \(M\neq \emptyset\) be a bounded \(F_ \sigma\) set. If the upper right density of \(M\) at each \(x\in M\) is \(\geq c\), then there exists \(y\not\in M\) at which the lower left density of \(M\) is \(\geq c\). This improves results of R. J. O'Malley, C. Freiling and P. Humke. 2. The analogous result does not hold for \(G_ \delta\) sets: there exists a \(G_ \delta\) set \(M\) such that the upper right density of \(M\) at each \(x\in M\) is equal to 1 and the lower left density of \(M\) is \(<1\) at each \(y\not\in M\).