The Hausdorff measure of a class of generalized Sierpiński sponges (Q2725218)

In this note, the Hausdorff measure of a class of generalized Sierpiński sponges is determined. The main result is the NEWLINENEWLINENEWLINETheorem: We have NEWLINE\[NEWLINEH^{s(\lambda)}(S^m(\lambda))= (\sqrt m)^{s(\lambda)},\quad m\geq 3,\quad 0< \lambda\leq 2^{-m},NEWLINE\]NEWLINE where \(S^m(\lambda)= \bigcap^\infty_{n=0} E^m_n(\lambda)\) is the so-called generalized Sierpiński sponge (if \(m=2\), \(\lambda= {1\over 4}\), then it is the Sierpiński carpet), and \(s(\lambda)= \dim_{\text{H}} S^m(\lambda)\) is the Hausdorff dimension.NEWLINENEWLINENEWLINEThe proof is based on 9 lemmas and the mass distribution principle.