HAUSDORFF MEASURE OF CARTESIAN PRODUCT OF THE TERNARY CANTOR SET

Abstract

Computing the Hausdorff measure of C × C, where C is the classical ternary Cantor set, is a long standing difficult problem. It is well-known that for a self-similar set, calculating the Hausdorff measure is equivalent to determining its optimal sets. This paper studies optimal sets of C × C: their diameters, measures, symmetries and the shapes. For this purpose, we introduce several devices: the repulsive principle, a bipartite graph G and a W-function. We show that the diameter of the optimal set B is between 1.2993 and 1.3082. Two symmetry properties of B are proved. Finally, we show that the shape of B is very close to a disk. We conjecture that an optimal set might be a disk.