Let E be a closed set on the unit circle. We find a Blaschke-type condition, optimal in a sense of the order, on the Riesz measure of a subharmonic function v in the unit disk with a certain growth at the direction of E. In particular case when E is a finite set, and v = log |f | with an analytic function f , our result agrees with the recent one by A. Borichev, L. Golinskii and S. Kupin. An application to contractions close to unitary operators in the Hilbert space is given.1991 Mathematics Subject Classification. Primary: 30D50; Secondary: 31A05, 47B10.Key words and phrases. analytic function in the unit disk, subharmonic function in the unit disk, Riesz measure, Blaschke-type condition, discrete spectrum, contraction operator.

We extend the notion of amoeba to holomorphic almost periodic functions in tube domains. In this setting, the order of a function in a connected component of the complement to its amoeba is just the mean motion of this function. We also find a correlation between the orders in different components.

J.C.Lagarias (2000) conjectured that if µ is a complex measure on pdimensional Euclidean space with a uniformly discrete support and its spectrum (Fourier transform) is also a measure with a uniformly discrete support, then the support of µ is a subset of a finite union of shifts of some full-rank lattice. The conjecture was proved by N. Lev and A.Olevski (2013) in the case p=1. In the case of an arbitrary p they proved the conjecture only for a positive measure µ.Here we show that Lagarias' conjecture is false in general case and find two new special cases when assertion of the conjecture is valid.

Abstract. We find new simple conditions for support of a discrete measure on Euclidean space to be a finite union of translated lattices. The arguments are based on a local analog of Wiener's Theorem on absolutely convergent trigonometric series and theory of almost periodic functions.

We continue the study of analytic functions in the unit disk of finite order with arbitrary set of singular points on the unit circle, introduced in [4]. The main focus here is made upon the inverse problem: the existence of a function from this class with a given singular set and zero set subject to certain Blaschke-type condition. We also discuss the local analog of the main result from [4] similar to the standard local Blaschke condition for analytic and bounded functions in the unit disk.