A new estimate of the Hausdorff measure of the Sierpinski gasket

Abstract

In this paper, we develop the hexagon method and the dodecagon method to estimate the Hausdorff measure of the Sierpiński gasket and show that the Hausdorff measure of the Sierpiński gasket is upper-bounded by a single-variable continuous function. Better upper bounds of the Hausdorff measure of the Sierpiński gasket are also achieved.

Bounds of the Hausdorff Measure of Sierpinski carpet

“…Like the Case (1), it is easy to prove that a (5) n decreases. Suppose that lim n→∞ a (5) n = α 5 ,by…”

Maximum density for the Sierpinski gasket

“…In [4], Marion gave an upper bound for s-dimensional Hausdorff measure H s (S) of the Sierpinski gasket S, H s (S) 3 s /6 ≈ 0.9508, and conjectured that this was its actual value of Hausdorff measure. But, Zhou [5] showed that the conjecture was not true and in [7] …”

Some remarks on the Hausdorff measure of the Cantor set

“…With the help of the principle of mass distribution, the Hausdorff measure of a Sierpinski carpet which equals 2 is obtained in [4]. The references [5] and [6] respectively gave the upper and lower bound for the Hausdorff measure of the Sierpinski gasket.…”