Large null sets in metric spaces (Q1773300)

Let \((X,\rho )\) be a locally compact \(\sigma \)-compact metric space of Hausdorff dimension \(\dim X = d > 0\) and let \(\mu \) be a Borel measure on \(X\) such that there exist constants \(0 < m \leq M\) such that for each \(x\in X\) and \(r\in (0,\operatorname{diam} X)\), \(mr^d \leq \mu (B(x,r)) \leq Mr^d\), (\(B(x,r)\) denotes the open ball with radius \(r\) centered at \(x\)). For a base system \((D_n)\) in \(X\) and \(\alpha > 0\) put \[ \begin{alignedat}{2} S(\alpha ) &= \bigcap_{k=1}^{\infty }\bigcup_{n=k}^{\infty } \biggl(\bigcup_{x\in D_n}B(x,n^{-\alpha (d+1)}) \biggr), &\quad K(\alpha ) &= \bigcap_{\beta < \alpha }S(\beta ),\\ L(\alpha )&= K(\alpha ) \,\Bigl\backslash\, \bigcup_{\beta > \alpha }S(\beta ),\qquad \text{ and} &L(\infty )&= \bigcap_{m\in N}S(m). \end{alignedat} \] It is proved that \(L(\infty )\) is residual in \(X\), \(\mu (\bigcup_{\alpha > \frac{1}{d}}S(\alpha )) = 0\) and \(\dim L(\alpha ) = \frac{1}{\alpha }\).