Dimensions of overlaps of self-similar fractals and self-similar multifractals

Let S-i : R-d --> R-d for i = 1,..., n be contracting similarities, and let (P-1, P-n) be a probability vector. Let K and mu be the self-similar set and the self-similar measure associated with (S-i, p(i))(j). Assume that the list (S-i)(i) satisfies the Open Set Condition and let t = (K) denote the Hausdorff dimension of K. It is well-known that the overlap SiK boolean AND SjK has zero t-dimensional Hausdorff measure for all i not equal j, i.e.,

(1) H-t(SiK boolean AND SjK) = 0 for all i not equal j,

where H-t denotes the t-dimensional Hausdorff measure. We improve this result by proving that the upper box-dimension, (dim) over bar (B) (SiK boolean AND SjK), of the overlap is strictly less than t = dim (K) for all i not equal j, i.e.,

(2) (dim) over bar (B)(SiK boolean AND SjK) <dim(K) for all i not equal j.

In fact, we prove a more general result saying that the overlaps SjK boolean AND SjK are also negligible from a multifractal point of view. We prove that for each q is an element of R, the upper multifractal q-box-dimension, (dim) over bar (q)(Bmu,B) (SiK boolean AND SjK), of the overlap is strictly less than the common value of the q-multifractal Hausdorff dimension, dim(mu)(q) (K), of K, and the L-q-spectrum tau(q) of mu, i.e.,

(3) (dim) over bar (B)dim(mu,B)(q) (SiK boolean AND SjK) < tau (q) = dim(mu)(q) (K) for all i not equal j. Observe that (2) follows from (3) by putting q = 0.