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Abstract. A self-affine set in R n is a compact set T with A(T ) = d∈D (T +d) whereA is an expanding n × n matrix with integer entries andis an N -digit set. For the case N = |det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > |det(A)|, but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T + d, d ∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T o = ∅). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.

For a d×d expanding matrix A, we define a pseudo-norm w(x) in terms of A and use this pseudo-norm (instead of the Euclidean norm) to define the Hausdorff measure and the Hausdorff dimension dim w H E for subsets E in R d . We show that this new approach gives convenient estimations to the classical Hausdorff dimension dimH E, and in the case that the eigenvalues of A have the same modulus, then dim w H E and dimH E coincide. This setup is particularly useful to study self-affine sets T generated by φj (x) = A −1 (x + dj), dj ∈ R d , j = 1, . . . , N. We use it to investigate the fractality of T for the case that {φj } N j=1 satisfying the open set condition as well as the cases without the open set condition. We extend some well-known results in the self-similar sets to the self-affine sets.

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