Heterogeneous ubiquitous systems in ℝ^d and Hausdorff dimension

Abstract.

Let {x _n }_n∈ℕ be a sequence in [0, 1]^d , {λ _n }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

$$ S{\left( \delta \right)} = {\bigcap\limits_{N \in \mathbb{N}} \; {{\bigcup\limits_{n \geqslant N} {B{\left( {x_{n} ,\lambda ^{\delta }_{n} } \right)}} }} }. $$

Let μ be a positive Borel measure on [0, 1]d , ρ 2 (0, 1] and α > 0. Consider the finer limsup-set

$$ S_{\mu } {\left( {p,\delta ,\alpha } \right)} = {\bigcap\limits_{N \in \mathbb{N}} \; {{\bigcup\limits_{n \geqslant N:\mu {\left( {B{\left( {x_{n} ,\lambda ^{p}_{n} } \right)}} \right)} \sim \lambda ^{{p\alpha }}_{n} } {B{\left( {x_{n} ,\lambda ^{\delta }_{n} } \right)}} }} }. $$

We show that, under suitable assumptions on the measure μ, the Hausdorff dimension of the sets S _μ (ρ, δ, α) can be computed. Moreover, when ρ < 1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension of S _μ (ρ, δ, α). Our results apply to several classes of multifractal measures, and S(δ) corresponds to the special case where μ is a monofractal measure like the Lebesgue measure.

The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with) b-adic expansion properties, by averages of some Birkhoff sums and branching randomwalks, as well as by asymptotic behavior of random covering numbers.