Measure of Self-Affine Sets and Associated Densities

Let (B) be an (ntimes n) real expanding matrix and (mathcal {D}) be a finite subset of (mathbb {R}^n) with (0in mathcal {D}) . The self-affine set (K=K(B,mathcal {D})) is the unique compact set satisfying the set-valued equation (BK=bigcup _{din mathcal {D}}(K+d)) . In the case where (#mathcal D=|det B|,) we relate the Lebesgue measure of (K(B,mathcal {D})) to the upper Beurling density of the associated measure (mu =lim _{srightarrow infty }sum _{ell _0, ldots ,ell _{s-1}in mathcal {D}}delta _{ell _0+Bell _1+cdots +B^{s-1}ell _{s-1}}.) If, on the other hand, (#mathcal D<|det B|) and (B) is a similarity matrix, we relate the Hausdorff measure (mathcal {H}^s(K)) , where (s) is the similarity dimension of (K) , to a corresponding notion of upper density for the measure (mu ) .