Sums of finite sets, orbits of commutative semigroups, and Hilbert functions (Q1910983)

This paper takes as a starting point Hilbert's theorem that the dimensions of the \(n\)-th filtered subspace of a graded module over a commutative polynomial algebra is given by a polynomial in \(n\), for \(n\) large enough. This is seen to be a special case of results about the size of finite subsets \(B + nA\) of a commutative semigroup \(G\), where \(B\) and \(A\) are fixed and \(n\) varies; and this in turn is a special case of the central theorem of the paper, which considers the sum of the values of a multiplicative character \(\chi : G \to \mathbb{C}\) over a subset \(B + nA\), and shows it to be given by an expression of the form \(\sum q^n_i P_i (n)\) for large \(n\). Here, the \(q_i\) are the values which \(\chi\) takes on \(A\), and \(P_i\) is a polynomial of degree less than the number of points of \(A\) at which \(\chi\) takes the value \(q_i\). On the way to proving this the author studies interesting ideas concerning the orbits of the semigroup \((\mathbb{N}, +)^{(t)}\) acting on a set, and makes crucial use of the classical fact that this semigroup is Noetherian. The present paper is a sequel to a paper by the author [Funct. Anal. Appl. 26, No. 4, 276-281 (1992); translation from Funkts. Anal. Prilozh. 26, No. 4, 57-63 (1992; Zbl 0809.13012)].