F_p is locally like C

TL;DR

This paper demonstrates that small subsets of finite fields can be embedded into algebraic extensions of Q while preserving algebraic relations, leading to improved combinatorial bounds and answering a question of Vu, Wood, and Wood.

Contribution

It establishes a converse to a known result, showing small subsets of F_p can be mapped into algebraic extensions of Q with preserved relations, with applications to combinatorics and number theory.

Findings

Szemerédi-Trotter theorem holds with optimal exponent 4/3 for small subsets of F_p

Improved sum-product estimates in F_p

Answered a question of Vu, Wood, and Wood regarding algebraic mappings

Abstract

Vu, Wood and Wood showed that any finite set S in a characteristic zero integral domain can be mapped to F_p, for infinitely many primes p, while preserving finitely many algebraic incidences of S. In this note we show that the converse essentially holds, namely any small subset of F_p can be mapped to some finite algebraic extension of Q, while preserving bounded algebraic relations. This answers a question of Vu, Wood and Wood. We give several applications, in particular we show that for small subsets of F_p, the Szemer\'edi-Trotter theorem holds with optimal exponent 4/3, and we improve the previously best-known sum-product estimate in F_p. We also give an application to an old question of R\'enyi. The proof of the main result is an application of elimination theory and is similar in spirit with the proof of the quantitative Hilbert Nullstellensatz.