On some counting problems for semi-linear sets

TL;DR

This paper proves that the growth functions of semi-linear sets in multi-dimensional integer spaces are box splines, providing a new proof of a classical theorem on counting solutions to linear Diophantine equations.

Contribution

It establishes that semi-linear set growth functions are box splines and offers a novel proof of a key theorem on counting solutions to linear Diophantine systems.

Findings

Growth functions of semi-linear sets are box splines.

Provides a new proof of Dahmen and Micchelli's theorem.

Connects semi-linear sets with spline theory.

Abstract

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.