N ov 2 00 7 Mapping Incidences
نویسندگان
چکیده
We show that any finite system S in a characteristic zero integral domain can be mapped to Z/pZ, 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, which asserts that any finite subset of a torsion-free group can be mapped into Z/pZ, preserving all linear incidences. As applications, we derive several combinatorial results (such as sum-product estimates) for a finite set in a characteristic zero integral domain. As C is a characteristic zero integral domain, this allows us to obtain new proofs for some recent results concerning finite sets of complex numbers, without relying on the topology of the plane.
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