A generalized \(\sigma\)-porous set with a small complement (Q2495020)

Let \(B(x,r)\) denote the ball with center~\(x\) and radius~\(r\) in a~metric space~\(X\). Let \(g:[0,\infty)\to[0,\infty)\) be a~continuous increasing function and let \(g(0)=0\). A~set \(M\subset X\) is called \(g\)-porous if for any \(x\in M\), \(\limsup_{s\to0}g(\gamma(x,s,M))/s>0\), where \(\gamma(x,s,M)=\sup\{r\geq0:\exists y\) \(B(y,r)\cap M=\emptyset\) and dist\((x,y)\leq s\}\). A~set which is a~countable union of \(g\)-porous sets is called a~\(\sigma\)-\(g\)-porous set. The main result of the paper says that if \(X\)~is a~Banach space and \(\limsup_{s\to0}g(s)/s=\infty\), then there is a~\(\sigma\)-\(g\)-porous set \(M\subseteq X\) such that the complement \(X\setminus M\) meets every \(C^1\)~curve in a~set of one-dimensional Hausdorff measure zero.