# A sum-product estimate in finite fields, and applications

@article{Bourgain2003ASE,
title={A sum-product estimate in finite fields, and applications},
author={Jean Bourgain and Nets Hawk Katz and Terence Tao},
journal={Geometric \& Functional Analysis GAFA},
year={2003},
volume={14},
pages={27-57}
}
• Published 29 January 2003
• Mathematics, Computer Science
• Geometric & Functional Analysis GAFA
AbstractLet A be a subset of a finite field $$F := \mathbf{Z}/q\mathbf{Z}$$ for some prime q. If $$|F|^{\delta} < |A| < |F|^{1-\delta}$$ for some δ > 0, then we prove the estimate $$|A + A| + |A \cdot A| \geq c(\delta)|A|^{1+\varepsilon}$$ for some ε = ε(δ) > 0. This is a finite field analogue of a result of [ErS]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as…
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## References

SHOWING 1-10 OF 41 REFERENCES
A new bound for finite field besicovitch sets in four dimensions
Let F be a finite field with characteristic greater than two. A Besicovitch set in F 4 is a set P ⊆ F 4 containing a line in every direction. The Kakeya conjecture asserts that P and F 4 have roughly
Freiman’s theorem concerns the structure of sets with small sumset. Let A be a subset of an abelian group G, and define the sumset A + A to be the set of all pairwise sums a + a′, where a, a′ are
On the number of sums and products
In what follows A will always denote a finite subset of the non-zero reals, and n the number of its elements. As usual, A + A and A · A stand for the sets of all pairwise sums {a + a : a, a ∈ A} and
A polynomial bound in Freiman's theorem
.Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s
On sums and products of integers
Erdős and Szemerédi conjectured that if A is a set of k positive integers, then there must be at least k2−ε integers that can be written as the sum or product of two elements of A. Erdős and
An analog of Freiman's theorem in groups
It is proved that any set A in a commutative group G where the order of elements is bounded by an integer r having n elements and at most n sums is contained in a subgroup of size An with A = f(r; )
Additive Number Theory: Inverse Problems and the Geometry of Sumsets
Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA
Extremal problems in discrete geometry
• Mathematics
Comb.
• 1983
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$
• Mathematics
• 2000
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorfi dimensions of such a set must be greater than or equal to 5= 2i n 3 . In
An improved bound on the Minkowski dimension of Besicovitch sets in R^3
• Mathematics
• 1999
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In