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Let $K$ be the attractor of a linear iterated function system (IFS)$S_j(x)=\rho_jx+b_j,$ $j=1,\cdots,m$, on the real line satisfying thegeneralized finite type condition (whose invariant open set $\mathcal{O}$ is aninterval) with an irreducible weighted incidence matrix. This condition wasintroduced by Lau \& Ngai recently as a natural generalization of the open setcondition, allowing us to include many important overlapping cases. They showedthat the Hausdorff and packing dimensions of $K$ coincide and can be calculatedin terms of the spectral radius of the weighted incidence matrix. Let $\alpha$be the dimension of $K$. In this paper, we state that \begin{equation*}\mathcal{H}^\alpha(K\cap J)\leq |J|^\alpha \end{equation*} for all intervals$J\subset\overline{\mathcal{O}}$, and \begin{equation*}\mathcal{P}^\alpha(K\cap J)\geq |J|^\alpha \end{equation*} for all intervals$J\subset\overline{\mathcal{O}}$ centered in $K$, where $\mathcal{H}^\alpha$denotes the $\alpha$-dimensional Hausdorff measure and $\mathcal{P}^\alpha$denotes the $\alpha$-dimensional packing measure. This result extends a recentwork of Olsen where the open set condition is required. We use theseinequalities to obtain some precise density theorems for the Hausdorff andpacking measures of $K$. Moreover, using these densities theorems, we describea scheme for computing $\mathcal{H}^\alpha(K)$ exactly as the minimum of afinite set of elementary functions of the parameters of the IFS. We also obtainan exact algorithm for computing $\mathcal{P}^\alpha(K)$ as the maximum ofanother finite set of elementary functions of the parameters of the IFS. Theseresults extend previous ones by Ayer \& Strichartz and Feng, respectively, andapply to some new classes allowing us to include linear Cantor sets withoverlaps.

Exact Hausdorff and packing measures of linear Cantor sets with overlaps