A Better Upper Bound of Hausdorff Measure of the Cartesian Product of the Middle Third Cantor Set with Itself Compare to Others'

Abstract

After the correction of an inaccurate result in the reference, the author uses five different methods, and gets five different inequalities on the Hausdorff measure of the Cartesian product of the middle third Cantor set with itself: $$H^s (C\times C)\leq 1.548563$$ $$H^s (C\times C)\leq 1.504975$$ $$H^s (C\times C)\leq 1.502878$$ $$H^s (C\times C)\leq 1.503263$$ $$H^s (C\times C)\leq 1.502483.$$ The main theory of this paper is the best among them, which is $$H^s (C\times C)\leq 1.502483.$$