The upper estimate and conjecture on Hausdorff measure of Sierpinski gasket
Xinghua Wang · 1999
For a binary decimal x with n bits, the pre-fracial of the level n + 1 of Sierpinski gasket S can be partially covered by a curved nonagon with the diameter 1 - x/2. The number of the small iriangles which are not com-pletely covered can be counted and a simple expression of the upper estimate function vn(x) for Hausdorff measure of S can be derived. By minimized searching, a sequence with optimized estimates is obtained. The values and expressions of the first 16 terms are given, especially, the 16th term is exp(ln 97793. ln 3/ln 2)799250914 = 0.817 930 0..... On the above basis, a conjecture on the exact value of Hausdorff measure is proposed.