Hausdorff measure of Sierpiński gasket (Q1387488)

The author gives a new estimate on the upper bound of the Hausdorff measure of the Sierpiński gasket \(S: H^s(S)\leq{25\over 22}\left({6\over 7}\right)^s\), where \(s= \log_23\) is the Hausdorff dimension of \(S\). The result improves the previous estimates obtained by the author [Proc. Nat. Sci. (English Ed.) 7, No. 4, 401-406 (1997)] and others [\textit{J. Marion}, Ann. Sci. Math. Qué. 11, 111-132 (1987; Zbl 0624.28003)]. In the proof, the author constructs a sequence of finite covers of \(S\), which gives rise to a descending sequence of the upper limits of the Hausdorff measure of \(S\). The limit of this descending sequence is \({25\over 22}\left({6\over 7}\right)^s\). Since the limit is again an upper bound of the Hausdorff measure of \(S\), the estimate is established.