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) = N∈N 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 Sµ(ρ, δ, α) = N∈N n≥N: µ(B(xn ,λ ρ n ))∼λ ρα n B(xn, λ δ n ).We show that, under suitable assumptions on the measure µ, the Hausdorff dimension of the sets Sµ(ρ, δ, α) can be computed. Moreover, when ρ < 1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension of Sµ(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 random walks, as well as by asymptotic behavior of random covering numbers.
This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ 2 beyond the transition phase (i.e. γ 2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.
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ABSTRACT. In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined bywhere µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2.The singularities analysis of the measures νγ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum of νγ,σ is linear on [0, hγ,σ] for some critical value hγ,σ, and then it is strictly concave on the right of hγ,σ, and deduced from the one of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. These measures open interesting perspectives in modeling discontinuous phenomena.
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