The concept of boundary layers, introduced by A. Volberg in [7], is generalized from subsets of the unit disk to subsets of general non-tangentially accessible (NTA) domains. Capacitary conditions of Wiener type series of both necessary and sufficient type for boundary layers are presented and the connection between boundary layers and minimally thin sets is studied.

We introduce a random recursive method for constructing random Salem sets in R^d. The method is inspired by Salem's construction [13] of certain signular monotonic functions.

We show that interpolation to a function, analytic on a compact set E in the complex plane, can yield maximal convergence only if a subsequence of the interpolation points converges to the equilibrium distribution on E in the weak sense. Furthermore, we will derive a converse theorem for the case when the measure associated with the interpolation points converges to a measure on E, which may be different from the equilibrium measure.

A new elementary approach to uniform resolvent estimates of the Carleman-type is developed. Schatten-von Neumann's $\mathfrak{S}_p $ perturbations of self-adjoint and unitary operators are considered. Examples of typical growth are provided.