Mapping Incidences

TL;DR

This paper demonstrates that finite sets in characteristic zero integral domains can be mapped to finite fields while preserving algebraic incidences, enabling new combinatorial applications like sum-product estimates.

Contribution

It generalizes the Freiman isomorphism lemma to characteristic zero integral domains and preserves incidences under mapping to finite fields.

Findings

Mapping of finite sets preserves algebraic incidences in finite fields

Enables combinatorial applications such as sum-product estimates

Works for infinitely many primes p

Abstract

We show that any finite set S in a characteristic zero integral domain can be mapped to the finite field of order p, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a generalization of the well-known Freiman isomorphism lemma, and we give several combinatorial applications (such as sum-product estimates).