We develop a new combinatorial method to estimate Hausdorff measures of various self-similar sets. This method can be applied to the evaluation of Hausdorff measures which are induced by various Hausdorff functions including power functions. Moreover, a few examples for evaluations of the lower and upper bounds of Hausdorff measures of uniform Cantor sets are introduced.
References
E. Best, On sets of fractal dimension III, Proc. London Math. Soc. (2) 47 (1942), 436–454. MR8421 10.1112/plms/s2-47.1.436 E. Best, On sets of fractal dimension III, Proc. London Math. Soc. (2) 47 (1942), 436–454. MR8421 10.1112/plms/s2-47.1.436
J. W. Fickett and J. Mycielski, A problem of invariance for Lebesgue measure, Colloq. Math. 42 (1979), 123–125. MR567553 0431.28002 J. W. Fickett and J. Mycielski, A problem of invariance for Lebesgue measure, Colloq. Math. 42 (1979), 123–125. MR567553 0431.28002
J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747. MR625600 0598.28011 10.1512/iumj.1981.30.30055 J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747. MR625600 0598.28011 10.1512/iumj.1981.30.30055
D. Kahnert, Haar-Mass und Hausdorff-Mass, Lecture Notes in Math. 541, Springer 1976, 13–23. MR450523 0364.28020 D. Kahnert, Haar-Mass und Hausdorff-Mass, Lecture Notes in Math. 541, Springer 1976, 13–23. MR450523 0364.28020
J. Mycielski, Remarks on invariant measures in metric spaces, Colloq. Math. 32 (1974), 109–116. MR361005 0298.28018 J. Mycielski, Remarks on invariant measures in metric spaces, Colloq. Math. 32 (1974), 109–116. MR361005 0298.28018