Sum-product Phenomenon Via Dimension

Yuyan He; Sergei Starchenko

Sum-product Phenomenon Via Dimension

Logic 2026-07-03 v1

Authors: Yuyan He , Sergei Starchenko

Abstract

We show a sum-product phenomenon in fields equipped with abstract dimension theories, which simultaneously generalizes the dimensions in geometric theories and Hrushovski's coarse pseudo-finite dimensions. More precisely, we show that for type-definable sets of positive non-zero dimension, non-expansion in dimension of both the sumset and product set implies the existence of a definable field in the same dimension. Main ingredients of the proof include dimensional analogues of the Ruzsa triangle inequality and the Pl\"unnecke-Ruzsa inequality.

Cite

@article{arxiv.2607.03384,
  title  = {Sum-product Phenomenon Via Dimension},
  author = {Yuyan He and Sergei Starchenko},
  journal= {arXiv preprint arXiv:2607.03384},
  year   = {2026}
}