The Integer Functions with Certain Arithmetic Properties

Integer functions the values of which have certain arithmetic properties at the points of a discrete set are studied. Given any nonnegative number p, we define the class F(p) consisting of all integer functions having the following properties: (a) The logarithm of the maximum modulus of each of the functions on a circle of radius R does not exceed R to power p (for sufficiently large values of R). (b) For each function of the class F(p) there is a sector with the center at zero, inside of which the function does not become zero at any point. (c) At the points of a complex two-dimensional lattice (of the general form), a function from the class F(p) takes a value from the ring of integers in a certain field of algebraic numbers K that is a finite extension of the field of rational numbers, and the logarithms of the heights of the function values at the lattice points lying inside the circle of radius R (centered at zero) do not exceed R to power p (for sufficiently large values of R). The structure of the classes F(p) with p lying in the interval with the ends equal to unity and the square root of two is described. It is shown that any function from this class is either a polynomial or is a rational function of a special form (the ratio of a polynomial to a monomial) from one or two exponentials with coefficients belonging to a certain field that is a finite extension of the field K. For obtaining the functional equations, Gelfond's classical method was used. In finding the integer solutions of the obtained functional equations, the authors used their newly developed technique of comparing close values of finite-order integer functions.