Almost periodic divisors in a strip S.Yu. Favorov, A.Yu. Rashkovskii and L.I.Ronkin 1 Introduction. Statement of the results The main goal of the paper is to find conditions for realizability of the so- called almost periodic divisors. Recall that a function f ∈ C (R) is almost periodic if for any ǫ> 0 the set E ǫ (f ):= {τ ∈ R : |f (t + τ ) − f (t)| < ǫ, ∀t ∈ R} - the set of ǫ-translations - is relatively dense in R. The latter means that for some L> 0,E ǫ (f ) ∩ (a, a + L)  = ∅, ∀a ∈ R. A notion of almost periodic function in a strip S := {z ∈ C : a< Im z<b}, −∞ ≤ a<b ≤∞, is defined similarly. Namely, a function f ∈ C (S ) is almost periodic if for any ǫ> 0 and any strip S 0 = {z ∈ C : α< Im z<β },a<α<β<b, the set E ǫ,S 0 (f ):= {τ ∈ R : |f (z + τ ) − f (z )| < ǫ, ∀z ∈ S 0 } is relatively dense in R. A typical example of a holomorphic almost periodic function in a strip S is a series of the form ∑ c n exp {iλ n z },λ n ∈ R, uniformly converging in the strip. In the sequel for the sake of brevity we write S 0 ⊂⊂ S in case of a< α<β<b. The space of functions holomorphic in a domain G is denoted by H (G). Throughout the paper z = x + iy, x ∈ R,y ∈ R. For a finite set E, Card E means the number of its elements. Continuous almost periodic functions on the axis (Bohr’s almost periodic functions) and holomorphic almost periodic functions in a strip were a subject of intensive study in the twenties-forties. Fundamental facts of the theory of almost periodic functions and its applications were found in papers of Bohr, Weyl, Wiener, von Neuman, Jessen, and others. 1