Metric, Fractal Dimensional and Baire Results on the Distribution of Subsequences

Abstract

Let X be a locally compact metric space. One important object connected with the distribution behavior of an arbitrary sequence x on X is the set M(x) of limit measures of x. It is defined as the set of accumulation points of the sequence of the discrete measures induced by x. Using binary representation of reals one gets a natural bijective correspondence between infinite subsets of the set ℕ of positive integers and numbers in the unit interval I = 〈0, 1]. Hence to each sequence x = (xn)n∈ℕ ∈ Xℕ and every a I there corresponds a subsequence denoted by ax. We investigate the set M(ax) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension.