The pointwise densities of the Cantor measure

[摘要] Let C be the classical middle-third Canter set and let mu (C) be the Canter measure. Set s = log2/log3. We will determine by an explicit formula for every point x epsilon C the upper and lower s-densities Theta* (s)(mu (C), x), Theta (s)* (mu (C), x) of the Canter measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F subset of C such that 9(Theta* (s)(mu (C), x))(-1/s) + (Theta (s)* (mu (C), x))(-1/s) = 16 holds for x epsilon C\F. Furthermore, for mu (C) almost all x, Theta*(s)(mu (C), x) = 2.4(-s) and Theta (s)* (mu (C), x) = 4(-s). As application, we will show that the s-dimensional packing measure of the middle-third Canter set C is 4(s). (C) 2000 Academic Press.

[发布日期] 2000-10-15

[关键词] Cantor measure;upper and lower density;packing measure