Zeroes of almost periodic analytic functions (Q2768851)

The following problem is studied: Under what conditions an almost periodic divisor is a set of zeroes of an almost periodic analytic function. Recall that a divisor in a domain \(G\subset\mathbb C\) is a set of pairs \(\{(a_n,k_n)\}\) where \(a_1, a_2,\ldots \) are isolated points from \(G\) with multiplicities \(k_1,k_2,\ldots \) respectively. A divisor \(d=\{(a_n,k_n)\}\) in a strip \(S:=\{z\in \mathbb C: a< \text{Im} z<b\}\) is said to be almost periodic if for any function \(\chi \in C(S \mapsto \mathbb R)\) with finite support the function \(t \mapsto \sum k_n\chi(a_n+t)\) is almost periodic. The union of spectra of all such almost periodic functions is called the spectrum of the divisor \(d\).NEWLINENEWLINENEWLINEThe author's approach in solving the problem uses cohomologies of Bohr compacts of additive groups. For an almost periodic divisor \(d\) with spectrum in an additive group \(\Gamma \), the author introduces a characteristic number \(c(d)\) as an element of the cohomology group \(H^2(K_\Gamma,\mathbb Z)\) and shows that \(c(d)=0\) iff \(d\) is a zero set of an almost periodic analytic function with spectrum in \(\Gamma \). He also establishes a number of important properties of the map \(d \mapsto c(d)\).