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

TL;DR

This paper refines the upper bound of the Hausdorff measure of the Cartesian product of the middle third Cantor set with itself, using five methods, and establishes the best inequality among them.

Contribution

The paper introduces a new, tighter upper bound for the Hausdorff measure of the Cantor set product, improving upon previous estimates.

Findings

Five different inequalities for the measure were derived.

The best upper bound found is 1.502483.

The new bound improves previous results.

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 $$H^s (C\times C)\leq 1.502483.$$