Combinatorial Bounds in Distal Structures

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• arXiv:2104.07769

• DOI: 10.1017/jsl.2023.74

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.

Citation

Aaron Anderson. “Combinatorial Bounds in Distal Structures.” Journal of Symbolic Logic (2023): 1-33.

@article{combdistal,
  title={Combinatorial bounds in distal structures},
  author={Anderson, Aaron},
  journal={The Journal of Symbolic Logic},
  volume={90},
  number={4},
  pages={1377--1409},
  year={2025},
  publisher={Cambridge University Press}
}