To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number-theoretical free-energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free-energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued-fraction digits exceeding a given number which tends to infinity.

We give a sufficient condition on a Lévy measure$μ$which ensures that the generator$L$of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., $\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu$ for all admissible$u$. In particular, we assume that $\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})$ . We also show that this condition is necessary provided that $\mathop {\mathrm {supp}}\mu$ is compact.

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Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and let$A(Δ)$be the algebra of complex functions continuous on Δ and analytic in int Δ. Let$K$be a compact set in C^{2}such that Π($K$)=Γ, and let$K$_{λ}≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,$K$_{λ}is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))^{2}-S(λ) quadratic in w with$R,S∈A(Δ)$such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ int$K$_{λ}, where$S$has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component of$K$_{λ}. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull of$K$. Furthermore, we show that БК/K is the disjoint union of such disks.