Parametrized Presburger Arithmetic: Logic, Combinatorics, and Quasi-polynomial Behavior

Parametric Presburger arithmetic concerns families of sets St ⊆ Zd, for t ∈ 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) ∈ Z[t]d, b(t) ∈ Z[t]. A function g : N → Z is a quasi-polynomial if there exists a period m and polynomials f0, . . . , fm−1 ∈ Q[t] such that g(t) = fi(t), for t ≡ 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, |St| might be an a quasi-polynomial function of t or an element x(t) ∈ St 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.