Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior
Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior, Discrete Analysis 2017:4, 34 pp.
Let \(T\) be a triangle with vertices \((0,0)\), \((0,1/3)\), and \((1,0)\), and let \(t\) be a positive integer. Then it is not hard to check that there are three quadratics \(q_1,q_2\) and \(q_3\) such that the number of integer points in \(tT\) is \(q_i(t)\) if \(t\equiv i\) mod 3. In a situation like this, we say that the number of integer points is quasipolynomial with period 3. In 1962, Ehrhart proved that if \(P\) is a polytope in \(\mathbb Z^d\) defined by a finite set of linear inequalities of the form \(a_i.x\leq b_i\), where each of the \(a_i\) belong to \(\mathbb Z^d\) and \(b\) belongs to \(\mathbb Z\), then the number of lattice points in \(tP\) is quasipolynomial with period \(m\), where \(m\) is the smallest integer such that the vertices of \(mP\) are all lattice points. Since then, the same conclusion has been established for other families of sets \(S_t\subset \mathbb Z^d\) by Chen, Li and Sam, by Calegari and Walker, and by Roune and Woods.
After these results, it was tempting to wonder whether all families of sets, provided that they are sufficiently nice in some appropriate sense, exhibit this quasipolynomial behaviour. The constraints would have to be reasonably strong – for example, the number of lattice points inside the unit sphere of radius \(t\) is certainly not quasipolynomial (and indeed, estimating it is a famous problem) – but one could still hope for a general theorem that would encompass the known results and give a number of further ones.
It turns out that the right behaviour to look for in general is that the size of \(S_t\) should be eventually quasipolynomial – that is, it should agree with a quasipolynomial for sufficiently large \(t\). Woods conjectured that eventual quasipolynomial behaviour should occur whenever the family is definable in parametric Presburger arithmetic. Roughly what this means (for a more precise definition, see the paper) is that the family \(S_t\) of subsets of \(\mathbb Z^d\) can be defined using addition, inequalities, integer constants, Boolean operations, multiplication by \(t\), and quantification over \(\mathbb Z\). The polytopes discussed earlier are examples. For a somewhat different kind of example, let \(S_t\) be the set of positive integers \(n\) such that there do not exist non-negative integers \(a,b,c\) with \(n=at+b(t+1)+c(t+2)\). This example involves quantification over \(\mathbb Z\), but again the number of points in \(S_t\) turns out to be quasipolynomial: in fact, it is \(\lfloor t^2/4 \rfloor\) (the paper also discusses a quasipolynomial formula for the maximum element of \(S_t\)). Note that it is crucial in this definition that multiplication, except by the parameter \(t\), should not be allowed, since otherwise we would have the full power of Peano arithmetic, which is undecidable.
The main result of this paper is a proof of this very appealing conjecture. The proof uses a series of reductions that make the family simpler and simpler until the result can be shown using previously developed methods. One of the reductions uses the well-known technique of quantifier elimination. However, this cannot be applied straightforwardly, owing to the multiplication-by-\(t\) operation, which is not part of standard Presburger arithmetic (hence the word “parametric”). The paper also discusses the power of parametric Presburger arithmetic, which, considering the necessary restrictions, is greater than one might expect. Thus, it proves eventual quasipolynomial behaviour for an extremely wide class of families and is probably the most general result one could hope for along these lines.