Combinatorial Bounds in Distal Structures

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• arXiv:2104.07769

• DOI: 10.1017/jsl.2023.74

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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. 2023. “Combinatorial Bounds in Distal Structures.” Accepted, Journal of Symbolic Logic.

@article{combdistal,
  title={Combinatorial Bounds in Distal Structures},
  DOI={10.1017/jsl.2023.74},
  journal={The Journal of Symbolic Logic},
  publisher={Cambridge University Press},
  author={Anderson, Aaron},
  year={2023},
  pages={1–29}
}