On some counting problems for semi-linear sets

Let $X$ be a subset of $\N^t$ or $\Z^t$. We can associate with $X$ a function ${\cal G}X:\N^t\longrightarrow\N$ which returns, for every $(n_1, …, n_t)\in \N^t$, the number ${\cal G}X(n_1, …, n_t)$ of all vectors $x\in X$ such that, for every $i=1,…, t, |x{i}| \leq n{i}$. This function is called the {\em growth function} of $X$. The main result of this paper is that the growth function of a semi-linear set of $\N^t$ or $\Z^t$ is a box spline. By using this result and some theorems on semi-linear sets, we give a new proof of combinatorial flavour of a well-known theorem by Dahmen and Micchelli on the counting function of a system of Diophantine linear equations.

💡 Research Summary

The paper investigates the counting function associated with semi‑linear subsets of the integer lattice ℕᵗ or ℤᵗ. For any such set X, the authors define the growth function 𝒢_X : ℕᵗ → ℕ by letting 𝒢_X(n₁,…,n_t) be the number of vectors x ∈ X whose coordinates satisfy |x_i| ≤ n_i for every i. This function measures, in a discrete sense, the “volume” of X inside an axis‑aligned box of side lengths (n₁,…,n_t).

The central theorem states that if X is semi‑linear—i.e., a finite union of linear (or affine) components of the form a + L where a ∈ ℤᵗ and L is a finitely generated sub‑lattice of ℤᵗ—then its growth function is a box spline. A box spline is a multivariate piecewise polynomial function whose domain is partitioned by a finite arrangement of hyperplanes determined by a set of direction vectors. Within each cell of this arrangement the function coincides with a polynomial; across cell boundaries the function is continuous and possesses a prescribed degree of smoothness.

To prove the theorem, the authors first treat a single affine component a + L. By counting lattice points in the intersection of a + L with the box