The Pointwise Densities of the Cantor Measure1

Let C be the classical middle-third Cantor set and let be the Cantor C measure. Set s log 2 log 3. We will determine by an explicit formula for every point x C the upper and lower s-densities sŽ , x., sŽ , x.of the C C Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F C such that 9Ž sŽ , x.. 1 s Ž s Ž , C C x.. 1 s 16 holds for x C F. Furthermore, for C almost all x, sŽ C , x. 2 4 sand sŽ , x. 4 s . As an application, we will show that the s-dimen- C sional packing measure of the middle-third Cantor set C is 4s.