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

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 = ( x n ) n ∈ℕ ∈ X ℕ and every a I there corresponds a subsequence denoted by a x . We investigate the set M ( a x ) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension.