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

Citations: | 50 - 4 self |

### BibTeX

@MISC{Bourgain_asum-product,

author = {Jean Bourgain and Nets Katz and Terence Tao},

title = {A sum-product estimate in finite fields, and applications},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If |F | δ < |A | < |F | 1−δ for some δ> 0, then we prove the estimate |A + A | + |A · A | ≥ c(δ)|A | 1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. 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 the three-dimensional Kakeya problem in finite fields. 1.

### Citations

1203 |
Modern Graph Theory
- Bollobas
- 1999
(Show Context)
Citation Context ...y 5.2 and the duality between points and lines in two dimensions we have the easy bounds |{(p, l) ∈ P × L : p ∈ l}| ≤ min(|P ||L| 1/2 + |L|, |L||P | 1/2 + |P |), (10)SUM-PRODUCT ESTIMATE 13 see e.g. =-=[Bol1978]-=-. In a sense, this is sharp: if we set N = |F | 2 , and let P be all the points in F 2 ⊂ PF 3 and L be most of the lines in F 2 , then we have roughly |F | 3 ∼ N 3/2 incidences. More generally if G is... |

139 | Additive Number Theory: Inverse Problems and the Geometry of - Nathanson - 1996 |

129 | On a set of distances of n points
- Erdős
- 1946
(Show Context)
Citation Context ... bound for |∆(P)| in terms of |P |. From the fact that any two “circles” intersect in at most two points, it is possible to use extremal graph theory to obtain the bound |∆(P)| ≥ c|P | 1/2 ; see also =-=[E1946]-=-. This bound is sharp if one takes P = F 2 , so that ∆(P) is essentially all of F. Similarly if one takes P = G 2 for any subfield G of F. However, as in the previous section one can hope to improve t... |

112 | A new proof of Szemerédi’s theorem for arithmetic progressions of length 4
- Gowers
- 1998
(Show Context)
Citation Context ...Then we have |A ± A ± A... ± A| ≤ CK C |A| for any additive combination of A, where the constants C depend on the length of this additive combination. Next, we recall Gowers’ quantitative formulation =-=[G1998]-=- of the Balog-Szemeredi lemma [BaSz1994]: Theorem 2.3. [G1998], [B1999] Let A, B be finite subsets of an additive group with cardinality |A| = |B|, and let G be a subset of A × B with cardinality such... |

87 |
The Geometry of Fractal Sets. Cambridge Tracts
- Falconer
- 1984
(Show Context)
Citation Context ...nown for some time that sum-product estimates have application to certain geometric combinatorics problems, such as the incidence problem for lines and the Erdös distance problem. (See e.g. [El1997], =-=[Fa1986]-=-, [ChuSzTr1992], [KT2001], [T.1]). Using these ideas (and particularly those from [KT2001], [T.1]), we can prove a theorem of Szemerédi-Trotter type in two-dimensional finite field geometries. The pre... |

64 | Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ
- Wolff
- 1996
(Show Context)
Citation Context ...every direction. The Kakeya conjecture for finite fields asserts that such sets have cardinality at least c(ε)q 3−ε for each ε > 0. The previous best lower bound known is cq 5/2 , and is due to Wolff =-=[W1999]-=- (see also [W1995], [MT]). We improve this to cq 5/2+ε for some absolute constant ε > 0. We prove this in Section 8, using some geometric ideas of the second author to transform the problem into a two... |

50 |
An improved bound for Kakeya type maximal functions
- Wolff
- 1995
(Show Context)
Citation Context ...he Kakeya conjecture for finite fields asserts that such sets have cardinality at least c(ε)q 3−ε for each ε > 0. The previous best lower bound known is cq 5/2 , and is due to Wolff [W1999] (see also =-=[W1995]-=-, [MT]). We improve this to cq 5/2+ε for some absolute constant ε > 0. We prove this in Section 8, using some geometric ideas of the second author to transform the problem into a twodimensional one, t... |

47 |
On the number of sums and products
- Elekes
- 1997
(Show Context)
Citation Context ...r problem, the analogue of (1) was obtained by Erdös and Szémeredi [ESz1983], with improvements in the value of ε by various authors; at present 1 the best known result is ε = 1/4, obtained by Elekes =-=[El1997]-=-. Also, a continuous version of (1), for fractal subsets of the real line, was recently obtained by the first author [B]. The proof of Theorem 1.1 is based on a recent argument of Edgar and Miller [Ed... |

44 | A polynomial bound in Freiman’s theorem
- Chang
(Show Context)
Citation Context ...ATZ, AND TERENCE TAO and a geometric progression simultaneously. This suggests using Freiman’s theorem [F1999] to obtain the estimate (1), but the best known quantitative bounds for Freiman’s theorem =-=[Cha2002]-=- are only able to gain a logarithmic factor in |A| over the trivial bound, as opposed to the polynomial gain of |A| ε in our result. We do not know what the optimal value of ε should be. If the finite... |

41 | On sums and products of integers
- Erdős, Szemerédi
- 1983
(Show Context)
Citation Context ...F := Z/qZ for some prime q. If |F | δ < |A| < |F | 1−δ for some δ > 0, then we prove the estimate |A + A| + |A · A| ≥ c(δ)|A| 1+ε for some ε = ε(δ) > 0. This is a finite field analogue of a result of =-=[ESz1983]-=-. 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 the three-dimensional Kake... |

36 |
On the dimension of Kakeya sets and related maximal inequalities
- Bourgain
- 1999
(Show Context)
Citation Context ... of A, where the constants C depend on the length of this additive combination. Next, we recall Gowers’ quantitative formulation [G1998] of the Balog-Szemeredi lemma [BaSz1994]: Theorem 2.3. [G1998], =-=[B1999]-=- Let A, B be finite subsets of an additive group with cardinality |A| = |B|, and let G be a subset of A × B with cardinality such that we have the bound |G| ≥ |A||B|/K |{a + b : (a, b) ∈ G}| ≤ K|A|. 2... |

26 | An improved bound on the Minkowski dimension of Besicovitch sets in R3
- Katz, Łaba, et al.
(Show Context)
Citation Context ..., to which the Szemerédi-Trotter theorem can then be applied. An analogous result in the continuous geometry R 3 will appear by the second author elsewhere. (An earlier result of Katz, ̷Laba, and Tao =-=[KLT2000]-=- also gives a similar result in the continuous case, but this result relies crucially on the fact that R has multiple scales, and so does not apply to the finite field problem). The third author is a ... |

23 |
A statistical theorem of set addition
- Balog, Szemerédi
- 1994
(Show Context)
Citation Context ...C |A| for any additive combination of A, where the constants C depend on the length of this additive combination. Next, we recall Gowers’ quantitative formulation [G1998] of the Balog-Szemeredi lemma =-=[BaSz1994]-=-: Theorem 2.3. [G1998], [B1999] Let A, B be finite subsets of an additive group with cardinality |A| = |B|, and let G be a subset of A × B with cardinality such that we have the bound |G| ≥ |A||B|/K |... |

18 |
An analog of Freiman’s theorem in groups, Structure theory of set addition, Astérisque No
- Ruzsa
- 1999
(Show Context)
Citation Context ... We need some notation. We say that a set A is essentially contained in B, and write A ⋐ B, if we have A ⊆ X + B for some set X of cardinality |X| ≤ CK C . We have the following simple lemma of Ruzsa =-=[R1999]-=-: Lemma 3.2. Let A and B be subsets of F such that |A+B| ≤ CK C |A| or |A−B| ≤ CK C |A|. Then B ⋐ A − A.SUM-PRODUCT ESTIMATE 7 Proof By symmetry we may assume that |A+B| ≤ CK C |A|. Let X be a maxima... |

16 | Tóth, Distinct distances in the plane
- Solymosi, Cs
(Show Context)
Citation Context ...n the Euclidean analogue to this problem, with N points in R 2 , it is conjectured [E1946] that the above estimate is true for all ε < 1/2. Currently, this is known4 for all ε < 4e 1 5e−1 − 2 ≈ 0.364 =-=[SoTaTo2001]-=-. However, the Euclidean results depend (among other things) on crossing number technology and thus do not seem to obviously extend to the finite field case. Proof We shall exploit the Szemerédi-Trott... |

13 |
Sums of finite sets, Number Theory: New York Seminar; SpringerVerlag
- Ruzsa
- 1996
(Show Context)
Citation Context ... | − 1 ≥ 1 2 |A||B| a,a ′ ∈A:a̸=a ′ b,b ′ ∈B:b̸=b ′ by our hypothesis |A||B| ≤ 1 2 |F |. The claim (3) then follows by the pigeonhole principle. We now recall the following sumset estimates (see e.g. =-=[R1996]-=-, [N1996]): Lemma 2.2 (Sumset estimates). Let A, B be a non-empty finite subsets of an additive group such that |A + B| ≤ K min(|A|, |B|). Then we have |A ± A ± A... ± A| ≤ CK C |A| for any additive c... |

11 |
jr., Extremal problems in discrete geometry
- Szemerédi, Trotter
- 1983
(Show Context)
Citation Context ...| ≤ CN 3/2−ε for some ε = ε(α) > 0 depending only on the exponent α. Remark. The corresponding statement for N points and N lines in the Euclidean plane R 2 (or PR 3 ) is due to Szeméredi and Trotter =-=[SzTr1983]-=-, with ε := 1/6. This bound is sharp. It may be that one could similarly take ε = 1/6 in the finite field case when α is sufficiently small, but we do not know how to do so; certainly the argument in ... |

9 |
A geometric inequality with applications to the Kakeya problem
- Schlag
- 1998
(Show Context)
Citation Context ...ell known that l lies in a quadratic surface 6 (i.e. a set of the form {x ∈ F 3 : Q(x) = 0} for some inhomogeneous quadratic polynomial Q) known as the regulus generated by l 1 , l 2 , l 3 ; see e.g. =-=[Sch1998]-=-, [T.2]. Thus in either case, all the lines l of interest lie inside an algebraic surface S which is either a plane or a quadratic surface. It will then suffice to show that the set has cardinality at... |

7 | Borel subrings of the reals
- Edgar, Miller
(Show Context)
Citation Context ...97]. Also, a continuous version of (1), for fractal subsets of the real line, was recently obtained by the first author [B]. The proof of Theorem 1.1 is based on a recent argument of Edgar and Miller =-=[EdMi2003]-=-, who solved the Erdös ring problem [EV1966]. (An alternate solution to this problem has also appeared in [B]). Specifically, these authors showed that there was no Borel subring A of the reals which ... |

7 |
Structure theory of set addition, Astèrisque 258
- Freiman
- 1999
(Show Context)
Citation Context ... as a statement that a set cannot behave like an arithmetic progression 12 JEAN BOURGAIN, NETS KATZ, AND TERENCE TAO and a geometric progression simultaneously. This suggests using Freiman’s theorem =-=[F1999]-=- to obtain the estimate (1), but the best known quantitative bounds for Freiman’s theorem [Cha2002] are only able to gain a logarithmic factor in |A| over the trivial bound, as opposed to the polynomi... |

7 | A new bound for finite field Besicovitch sets in four dimensions - Tao |

6 |
Additive Gruppen mit vorgegebener Hausdorffscher Dimension
- Erdős, Volkmann
- 1966
(Show Context)
Citation Context ...actal subsets of the real line, was recently obtained by the first author [B]. The proof of Theorem 1.1 is based on a recent argument of Edgar and Miller [EdMi2003], who solved the Erdös ring problem =-=[EV1966]-=-. (An alternate solution to this problem has also appeared in [B]). Specifically, these authors showed that there was no Borel subring A of the reals which has Hausdorff dimension strictly between 0 a... |

5 |
Kakeya and restriction phenomena for finite fields
- Mockenhaupt, Tao
(Show Context)
Citation Context ... conjecture for finite fields asserts that such sets have cardinality at least c(ε)q 3−ε for each ε > 0. The previous best lower bound known is cq 5/2 , and is due to Wolff [W1999] (see also [W1995], =-=[MT]-=-). We improve this to cq 5/2+ε for some absolute constant ε > 0. We prove this in Section 8, using some geometric ideas of the second author to transform the problem into a twodimensional one, to whic... |

4 |
Some connections between the Falconer and Furstenburg conjectures
- Katz, Tao
(Show Context)
Citation Context ...um-product estimates have application to certain geometric combinatorics problems, such as the incidence problem for lines and the Erdös distance problem. (See e.g. [El1997], [Fa1986], [ChuSzTr1992], =-=[KT2001]-=-, [T.1]). Using these ideas (and particularly those from [KT2001], [T.1]), we can prove a theorem of Szemerédi-Trotter type in two-dimensional finite field geometries. The precise statement is in Theo... |

2 |
On the number of different distances, Discrete and Computational Geometry 7
- Chung, Szemerédi, et al.
- 1992
(Show Context)
Citation Context ...ome time that sum-product estimates have application to certain geometric combinatorics problems, such as the incidence problem for lines and the Erdös distance problem. (See e.g. [El1997], [Fa1986], =-=[ChuSzTr1992]-=-, [KT2001], [T.1]). Using these ideas (and particularly those from [KT2001], [T.1]), we can prove a theorem of Szemerédi-Trotter type in two-dimensional finite field geometries. The precise statement ... |

1 |
On the Erdös ring problem, preprint
- Bourgain
(Show Context)
Citation Context ...authors; at present 1 the best known result is ε = 1/4, obtained by Elekes [El1997]. Also, a continuous version of (1), for fractal subsets of the real line, was recently obtained by the first author =-=[B]-=-. The proof of Theorem 1.1 is based on a recent argument of Edgar and Miller [EdMi2003], who solved the Erdös ring problem [EV1966]. (An alternate solution to this problem has also appeared in [B]). S... |

1 | Finite field analogues of the Erdös - Tao |