Let
$K$
be the attractor of a linear iterated function system (IFS)
$S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$
, on the real line
$\mathbb{R}$
satisfying the generalized finite type condition (whose invariant open set
${\mathcal{O}}$
is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of
$K$
coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let
${\it\alpha}$
be the dimension of
$K$
. In this paper, we state that
$$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$
for all intervals
$J\subset \overline{{\mathcal{O}}}$
, and
$$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$
for all intervals
$J\subset \overline{{\mathcal{O}}}$
centered in
$K$
, where
${\mathcal{H}}^{{\it\alpha}}$
denotes the
${\it\alpha}$
-dimensional Hausdorff measure and
${\mathcal{P}}^{{\it\alpha}}$
denotes the
${\it\alpha}$
-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets.
Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of
$K$
. Moreover, using these density theorems, we describe a scheme for computing
${\mathcal{H}}^{{\it\alpha}}(K)$
exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing
${\mathcal{P}}^{{\it\alpha}}(K)$
as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets.
Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets.
Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in
$\mathbb{R}$
with overlaps.