Analogues of the Lebesgue Density Theorem for Fractal Sets of Reals and Integers

We prove the following analogues of the Lebesgue density theorem for two types of fractal subsets of R: cookie-cutter Cantor sets and the zero set of a Brownian path. Write C for the set, and μ for the positive finite Hausdorff measure on C. Then there exists a constant c (depending on the set C) such that for μ-almost every x ∈ C, lim T → ∞ 1 T ∫ 0 T μ ( B ( x , e - t ) ) ( 2 e - t ) d d t = c where B(x, ε) is the ε-ball around x and d is the Hausdorff dimension of C. We also define analogues of Hausdorff dimension and Lebesgue density for subsets of the integers, and prove that a typical zero set of the simple random walk has dimension ½ and density √(2/π).