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Abstract. Let H be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every k ∈ N, let U k (H) denote the set of all ℓ ∈ N with the property that there are atomsis the union of all sets of lengths containing k).The Structure Theorem for Unions states that, for all sufficiently large k, the sets U k (H) are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds.This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

Let P(N) be the power set of N. We say that a function µ ⋆ : P(N) → R is an upper density if, for all X, Y ⊆ N and h, k ∈ N + , the following hold: (f1) µ ⋆ (N) = 1; (f2)We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya, and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)-(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that µ ⋆ (X) ≤ 1 for every X ⊆ N.Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

We introduce the sequence of generalized Gončarov polynomials, which is a basis for the solutions to the Gončarov interpolation problem with respect to a delta operator. Explicitly, a generalized Gončarov basis is a sequence (tn(x)) n≥0 of polynomials defined by the biorthogonality relation εz i (d i (tn(x))) = n! δi,n for all i, n ∈ N, where d is a delta operator, Z = (zi) i≥0 a sequence of scalars, and εz i the evaluation at zi. We present algebraic and analytic properties of generalized Gončarov polynomials and show that such polynomial sequences provide a natural algebraic tool for enumerating combinatorial structures with a linear constraint on their order statistics.2010 Mathematics Subject Classification. Primary 05A10, 41A05. Secondary 05A40.