A sum-product estimate in finite fields, and applications. (English) Zbl 1145.11306

Bourgain, Jean; Katz, N.; Tao, Terence C.

Geom. Funct. Anal. 14, No. 1, 27-57 (2004).

Summary: Let \(A\) be a subset of a finite field \(\mathbb F := \mathbb Z/q\mathbb Z\) for some prime \(q\). If \(|\mathbb F|^{\delta} < |A| < |\mathbb F|^{1-\delta}\) for some \(\delta > 0\), then we prove the estimate \(|A + A| + |A \cdot A| \geq c(\delta)|A|^{1+\varepsilon}\) for some \(\varepsilon= \varepsilon(\delta) > 0\). This is a finite field analogue of a result of P. Erdős and E. Szemerédi [Studies in Pure Mathematics, Mem. of P. Turán, 213–218 (1983; Zbl 0526.10011)]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdős distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.

MSC:

11B75	Other combinatorial number theory
11T30	Structure theory for finite fields and commutative rings (number-theoretic aspects)

Citations:

Zbl 0526.10011