Combinatorial Bounds in Distal Structures

Abstract

We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly o-minimal and P -minimal structures. The bound in general weakly o-minimal structures generalizes the vertical cell decomposition for semialgebraic sets, and the bounds for vector spaces in both o-minimal and p-adic cases are tight. We apply these bounds to Zarankiewicz's problem and sum-product bounds in distal structures. PreliminariesIn this section, we review the notation and model-theoretic framework necessary to understand distal cell decompositions. For further background on these definitions, see [4] and [5].Firstly, we review asymptotic notation:• We will say f (x) = Ω(g(x)) to indicate that there existsThroughout this section, let M be a first-order structure in the language L. We will frequently refer to Φ(x; y) as a set of formulas, which will implicitly be in the language L. Each formula in Φ will have the same variables, split into a tuple x and a tuple y, where, for instance, |x| represents the length of the tuple x. We use M to refer to the universe, or underlying set, of M, and M n to refer to its nth Cartesian power. If φ(x; y) is a formula with its variables partitioned into x and y, and b ∈ M |y| , then φ(M |x| ; b) refers to the definable set {a ∈ M |x| : M |= φ(a, b)}. We also define the dual formula of φ(x; y) to be φ * (y; x) such that M |= ∀x∀yφ(x; y) ↔ φ * (y; x), and similarly define Φ * (y; x) to be the set {φ * (y; x) : φ(x; y) ∈ Φ(x; y)}.