We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that if G is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1.

TheH^p corona problem is the following: Let g₁, ..., g_m be bounded holomorphic functions with 0<δ≤Σ‖g_i‖. Can we, for any H^p function ϕ, find H^p functions u₁, ..., u_m such that Σg_iu_i=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. The H^p estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

Let (X, L) be a polarized 3-fold over the complex number field. In [Fk3], we proved that g(L)≥q(X) if h⁰(L)≥2 and moreover we classified (X, L) with h⁰(L)≥3 and g(L)=q(X), where g(L) is the sectional genus of (X, L) and q(X)=dimH¹(O_X) the irregularity of X. In this paper we will classify polarized 3-folds (X, L) with h⁰(L)≥4 and g(L)=q(X)+1 by the method of [Fk3].

One of our main results is the following: Let X be a compact connected subset of the Euclidean space Rⁿ and r(X, d₂) the rendezvous number of X, where d₂ denotes the Euclidean distance in Rⁿ. (The rendezvous number r(X, d₂) is the unique positive real number with the property that for each positive integer n and for all (not necessarily distinct)x₁, x₂,..., x_n in X, there exists some x in X such that $(1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )$ .) Then there exists some regular Borel probability measure μ₀ on X such that the value of ∫_Xd₂(x, y)dμ₀ (y) is independent of the choice x in X, if and only if r(X, d₂) = sup_μ ∫_X ∫_Xd₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ on X.