Combinatorial Bounds in Distal Structures

TL;DR

This paper establishes polynomial upper bounds for the sizes of distal cell decompositions in various distal structures, extending known results and applying them to problems like Zarankiewicz's and sum-product bounds.

Contribution

It introduces new polynomial bounds for distal cell decompositions in weakly o-minimal and P-minimal structures, generalizing previous results and providing tight bounds for vector spaces.

Findings

Polynomial bounds for distal cell decompositions in weakly o-minimal structures

Tight bounds for vector spaces in o-minimal and p-adic cases

Applications to Zarankiewicz's problem and sum-product bounds

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.