Heterogeneous Ubiquitous Systems in R and Hausdorff Dimension

Let {xn}n∈N be a sequence of [0, 1]d, {λn}n∈N a sequence of positive real numbers converging to 0, and δ > 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the form S(δ) = T N∈N S n≥N B(xn, λ δ n). Let μ be a positive Borel measure on [0, 1]d, ρ ∈ (0, 1] and α > 0. Consider the finer limsup-set