We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

In a recent paper, Pertti Mattila asked which gauge functions φ have the property that for any Borel set A ⊂ R2 with Hausdorff measure Hφ(A) > 0, the projection of A to almost every line has positive length. We show that finiteness of ∫ 1 0 φ(r) r2 dr, which is known to be sufficient for this property, is also necessary for regularly varying φ. Our proof is based on a random construction adapte...

Let y = h(x) be defined for 0 < x < oo and assume values in 0 ^ y ^ + co. Let S be any linear set of points and p an arbitrary positive number. Cover S by a countable number of open intervals h , I2, • • • of lengths xx, x2 , • • • each of which is less than p, and denote by mp(S; h) the lower bound of h(xi) + h(x2) + • • • for all such coverings of S. Then m(S; h) = limp|0mp(/S; h) is called t...

The computation of the Hausdorff measure of fractals is the basic problem in fractal geometry. However, this is very difficult. The genetic algorithm is one of optimization algorithms to resolve complicated problems of wide scope, and has great capabilities in self-organizing, self-adaptation and self-learning. Lifeng Xi professor put forward to the thought of computing the Hausdorff measure of...