Mapping Incidences
arXiv:0711.4407 · doi:10.1112/jlms/jdr017
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).
15 pages, to appear in the Journal of the London Mathematical Society. Section 3 on Erdős distance problem from the previous version has been removed, since the most current version of arXiv:math/0301343v3 [math.CO] has the added restriction that -1 is not a square. Other minor revisions were also made