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.

References in corpus (1)

• Incidence bounds in positive characteristic via valuations and distality

Cited by in corpus (1)

• Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications