Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

TL;DR

This paper proves Woods' conjecture that all parametric Presburger sets exhibit quasi-polynomial behavior, using tools from logic and combinatorics, advancing understanding of the structure of these sets.

Contribution

The paper establishes that every parametric Presburger set displays quasi-polynomial behavior, confirming Woods' conjecture and providing a comprehensive characterization of these sets.

Findings

All parametric Presburger sets are quasi-polynomial.

The proof employs logic and combinatorics techniques.

Supports previous specific cases and conjectures.

Abstract

Parametric Presburger arithmetic concerns families of sets S_t in Z^d, for t in N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the form a(t) x <= b(t), where a(t) is in Z[t]^d, b(t) in Z[t]. A function g: N -> Z is a quasi-polynomial if there exists a period m and polynomials f_0, ..., f_{m-1} in Q[t] such that g(t)=f_i(t) for t congruent to i (mod m.) Recent results of Chen, Li, Sam; Calegari, Walker; Roune, Woods; and Shen concern specific families in parametric Presburger arithmetic that exhibit quasi-polynomial behavior. For example, S_t might be an a quasi-polynomial function of t or an element x(t) in S_t might be specifiable as a function with quasi-polynomial coordinates, for sufficiently large t. Woods conjectured that all parametric Presburger sets exhibit this quasi-polynomial behavior. Here, we prove this conjecture, using various tools from logic and combinatorics.