On the pointwise densities of the Cantor measure

Let K ( a ) be the symmetrical Cantor set generated by ϕ 0 ( x ) = a x and ϕ 1 ( x ) = a x + ( 1 − a ) , where 0 < a < 1 / 2 . Let s be the Hausdorff dimension of K ( a ) and μ the Cantor measure. In this paper, under the hypothesis that a is slightly greater than 1/3, we obtain the explicit formulas of the upper and lower s-densities Θ ⁎ s ( μ , x ) , Θ ⁎ s ( μ , x ) for every point x ∈ K ( a ) . Moreover, we describe the range of a key quantity τ ( x ) in these formulas.