## Finite field models in arithmetic combinatorics

Citations: | 8 - 1 self |

### BibTeX

@MISC{Green_finitefield,

author = {Ben Green},

title = {Finite field models in arithmetic combinatorics},

year = {}

}

### OpenURL

### Abstract

Abstract. The study of many problems in additive combinatorics, such as Szemerédi’s theorem on arithmetic progressions, is made easier by first studying models for the problem in Fn p, for some fixed small prime p. We give a number of examples of finite field models of this type, which allows us to introduce some of the central ideas in additive combinatorics relatively cleanly. We also give an indication of how the intuition gained from the study of finite field models can be helpful for addressing the original questions. 1.

### Citations

203 | Szemerédi’s regularity lemma and its applications in graph theory
- Komlós, Simonovits
- 1996
(Show Context)
Citation Context ... [13] later defined quasi-randomness for subsets of Z/NZ. Quasirandomness has been most thoroughly explored in the context of graphs, for which the reader should consult the excellent survey articles =-=[38, 39]-=-. The notions of uniformity in Z/NZ and in Fn p differ in little more than notation. As an example of uniformity/quasirandomness at work, and to get comfortable with the notation, let us prove that un... |

151 | The primes contain arbitrarily long arithmetic progressions, preprint: http://xxx.arxiv.org/math - Green, Tao |

143 | Regular partitions of graphs, in: Problèmes combinatoires et théorie des graphes - Szemerédi - 1978 |

140 |
On certain sets of integers
- Roth
- 1953
(Show Context)
Citation Context ...bset of {1, . . ., N} containing no three distinct elements x, x + d, x + 2d in arithmetic progression? This question was first raised by Erdős and Turán in 1936 [16], and was addressed by Klaus Roth =-=[46]-=-. Define r3(N) to be the answer to Problem 1.1. Roth proved that r3(N) ≪ N/ log log N, a bound which was improved to N(log N) −c independently by Heath-Brown [36] and Szemerédi [57], and then to r3(N)... |

134 | Additive Number Theory: Inverse Problems and Geometry of Sumsets, ser. Graduate Texts in Mathematics - Nathanson - 1996 |

102 |
The Probabilistic Method, 2nd ed
- Alon, Spencer
- 2000
(Show Context)
Citation Context ...y, by including each x ∈ F n 2 in A independently at random with probability 1/2) will be η-uniform with very high probability. In fact, using a large deviation estimate such as Chernoff’s bound (see =-=[5]-=- for example) one can show that this is true even4 BEN GREEN for η = N −1/2+ǫ . A truly random set will have many other properties almost surely. Remarkably, many of these are consequences of A being... |

100 | Additive Combinatorics - Tao, Vu - 2006 |

93 | The algorithmic aspects of the regularity lemma - Alon, Duke, et al. - 1994 |

86 |
Triple systems with no six points carrying three triangles
- Ruzsa, Szemerédi
- 1976
(Show Context)
Citation Context ...t C1(δ)N 3 triangles, for some C(δ) > 0. 3 We have not attributed this result, as it is not clear to us where it was first stated. A slightly weaker result was obtained by Ruzsa and Szemerédi in 1976 =-=[52]-=-. The result is also well-known in the literature concerning “property testing”: see, for example, [3].FINITE FIELD MODELS IN ARITHMETIC COMBINATORICS 15 Put another way, if a graph is almost triangl... |

82 | On some sequences of integers
- Erdős, Turán
- 1936
(Show Context)
Citation Context ...t is the cardinality of the largest subset of {1, . . ., N} containing no three distinct elements x, x + d, x + 2d in arithmetic progression? This question was first raised by Erdős and Turán in 1936 =-=[16]-=-, and was addressed by Klaus Roth [46]. Define r3(N) to be the answer to Problem 1.1. Roth proved that r3(N) ≪ N/ log log N, a bound which was improved to N(log N) −c independently by Heath-Brown [36]... |

65 |
Lower bounds of tower type for Szemerédi’s uniformity lemma, GAFA 7
- Gowers
- 1997
(Show Context)
Citation Context ...hat M(ǫ) grows like a tower of twos of height ǫ −3 . This is because each application of Lemma 7.4 results in an exponentiation of the codimension of H. By adapting a brilliant construction of Gowers =-=[20]-=-, which shows that Szemerédi’s regularity lemma for graphs must have tower type bounds, we were able to show that M(ǫ) must be at least as bad as a tower of twos of height about log(1/ǫ). We were not ... |

60 |
Foundations of a structural theory of set addition
- Freiman
- 1973
(Show Context)
Citation Context ...ontext of general abelian groups G. However, the issues are of a rather different nature to those discussed in §8. Freiman’s original work concerned subsets of Z, and was quite geometric in feel. See =-=[7, 17, 27]-=- for a further discussion. Ruzsa’s proof [48] has proved much more adaptable, and recently Ruzsa and the author [32] were able to obtain a structure theorem for sets with small doubling which is valid... |

50 |
Rough classification
- Pawlak
- 1984
(Show Context)
Citation Context ...lynomial in K. Ruzsa was probably the first to actually dare to conjecture this, and he certainly states such a conjecture explicitly in [51]. Such matters are also touched upon (in the Z-setting) in =-=[11, 24]-=-. Imre Ruzsa indicated to me a large part of the following proposition giving a number of statements equivalent to such a structure theorem. The proof may be found in [30]. Proposition 10.2 (Ruzsa). T... |

49 |
A.: Pseudorandom graphs. In: Random graphs ’85
- Thomason
- 1985
(Show Context)
Citation Context ... truly random set will have many other properties almost surely. Remarkably, many of these are consequences of A being η-uniform. This phenomenon was investigated in the context of graphs by Thomason =-=[62, 63]-=- and by Chung, Graham and Wilson [14]. Chung and Graham [13] later defined quasi-randomness for subsets of Z/NZ. Quasirandomness has been most thoroughly explored in the context of graphs, for which t... |

48 | Integer sets containing no arithmetic progressions
- Heath-Brown
- 1987
(Show Context)
Citation Context ...[16], and was addressed by Klaus Roth [46]. Define r3(N) to be the answer to Problem 1.1. Roth proved that r3(N) ≪ N/ log log N, a bound which was improved to N(log N) −c independently by Heath-Brown =-=[36]-=- and Szemerédi [57], and then to r3(N) ≪ N(log log N/ log N) 1/2 by Bourgain [10]. We are still a long way from a complete understanding of r3(N); the best known lower bound is Behrend’s [6] 1946 exam... |

43 | A polynomial bound in Freiman’s theorem
- Chang
(Show Context)
Citation Context ...s a subspace with codimension f(α). What is the behaviour of f(α)? Using a Fourier-analytic technique of Bogolyubov [8] one may show that f(α) ≪ α −2 , and a refinement of this technique due to Chang =-=[12]-=- allows one to improve this to f(α) ≪ α −1 log(1/α). We have not been able to rule out the possibility that f(α) ≪ log(1/α), which if true would imply PFR. The proof of Proposition 10.2 uses an import... |

30 |
On subsets of finite abelian groups with no 3-term arithmetic progressions
- Meshulam
- 1995
(Show Context)
Citation Context ...It turns out that both Problems 1.1 and 1.2 are both considerably easier in groups other than those in which they were originally asked (Z/NZ for Problem 1.11 and Z for Problem 1.2). Indeed, Meshulam =-=[41]-=- observed that Problem 1.1 is naturally addressed in Fn 3 , whereas Ruzsa [47] saw that Problem 1.2 is particularly pleasant in F∞ 2 . Here, Fp denotes the finite field with p elements, and F∞ p is ou... |

26 |
Testing subgraphs in large graphs, Random Structures Algorithms 21
- Alon
- 2002
(Show Context)
Citation Context ... where it was first stated. A slightly weaker result was obtained by Ruzsa and Szemerédi in 1976 [52]. The result is also well-known in the literature concerning “property testing”: see, for example, =-=[3]-=-.FINITE FIELD MODELS IN ARITHMETIC COMBINATORICS 15 Put another way, if a graph is almost triangle-free (i.e. contains few triangles) then it can be made truly triangle-free by the removal of a small... |

25 | The ergodic theoretical proof of Szemerédi’s theorem - Furstenberg, Katznelson, et al. - 1982 |

25 | Restriction and Kakeya phenomena for finite fields
- Mockenhaupt, Tao
(Show Context)
Citation Context ...atics where finite field models have proved invaluable, such as the study of the Kakeya and restriction phenomena. We do not touch upon these matters here, referring the reader instead to the article =-=[42]-=- as well as in the surveys [37, 58, 59]. 2. Notation and Basic Definitions Let p be a prime (p will be either 2,3 or 5). Write Fp for the finite field with p elements, which may be identified with Z/p... |

22 |
Quasi-random subsets of Zn
- Chung, Graham
- 1992
(Show Context)
Citation Context ... Remarkably, many of these are consequences of A being η-uniform. This phenomenon was investigated in the context of graphs by Thomason [62, 63] and by Chung, Graham and Wilson [14]. Chung and Graham =-=[13]-=- later defined quasi-randomness for subsets of Z/NZ. Quasirandomness has been most thoroughly explored in the context of graphs, for which the reader should consult the excellent survey articles [38, ... |

18 |
An analog of Freiman’s theorem in groups, Structure theory of set addition, Astérisque No
- Ruzsa
- 1999
(Show Context)
Citation Context ...oups other than those in which they were originally asked (Z/NZ for Problem 1.11 and Z for Problem 1.2). Indeed, Meshulam [41] observed that Problem 1.1 is naturally addressed in Fn 3 , whereas Ruzsa =-=[47]-=- saw that Problem 1.2 is particularly pleasant in F∞ 2 . Here, Fp denotes the finite field with p elements, and F∞ p is our notation for a vector space of countable dimension over Fp. Roughly speaking... |

15 |
graphs, strongly regular graphs and pseudorandom graphs, Surveys in Combinatorics
- Random
- 1987
(Show Context)
Citation Context ... truly random set will have many other properties almost surely. Remarkably, many of these are consequences of A being η-uniform. This phenomenon was investigated in the context of graphs by Thomason =-=[62, 63]-=- and by Chung, Graham and Wilson [14]. Chung and Graham [13] later defined quasi-randomness for subsets of Z/NZ. Quasirandomness has been most thoroughly explored in the context of graphs, for which t... |

14 |
Sets of lattice points that form no squares
- Ajtai, Szemerédi
- 1974
(Show Context)
Citation Context ...n of Problem 1.1: Problem 5.1. What is r∠(N), the cardinality of the largest subset of {1, . . ., N} × {1, . . ., N} containing no corner ((x, y), (x + d, y), (x, y + d)), d ̸= 0? Ajtai and Szemerédi =-=[2]-=- proved that r∠(N) = o(N), and various subsequent authors [54, 64] have obtained explicit bounds of the shape r∠(N) ≪ N/(log ∗ N) c . Here log ∗ N is the number of times one must take the logarithm of... |

13 | A simple algorithm for constructing Szemerédi’s regularity partition
- Frieze, Kannan
- 1999
(Show Context)
Citation Context ...e are large vertex sets B1, B2 such that the edge density of ΓA restricted to B1 ∪ B2 is much greater than α. Shkredov in effect provides a spectral proof of this statement, which is in the spirit of =-=[18]-=-. A purely combinatorial proof is more traditional, and somewhat simpler – the details may be found in [29].10 BEN GREEN The discussion of the last paragraph might suggest that we should enlarge Stru... |

13 |
arithmetical progressions and sumsets
- Generalized
- 1994
(Show Context)
Citation Context ...ues are of a rather different nature to those discussed in §8. Freiman’s original work concerned subsets of Z, and was quite geometric in feel. See [7, 17, 27] for a further discussion. Ruzsa’s proof =-=[48]-=- has proved much more adaptable, and recently Ruzsa and the author [32] were able to obtain a structure theorem for sets with small doubling which is valid in any abelian group. Theorem 10.7 (G. – Ruz... |

12 |
Structure of sets with small sumset
- Bilu
- 1999
(Show Context)
Citation Context ...ontext of general abelian groups G. However, the issues are of a rather different nature to those discussed in §8. Freiman’s original work concerned subsets of Z, and was quite geometric in feel. See =-=[7, 17, 27]-=- for a further discussion. Ruzsa’s proof [48] has proved much more adaptable, and recently Ruzsa and the author [32] were able to obtain a structure theorem for sets with small doubling which is valid... |

12 |
Note on a generalization of Roth’s theorem,Discrete and computational geometry, 825–827, Algorithms Combin. 25
- Solymosi
- 2003
(Show Context)
Citation Context ... of the largest subset of {1, . . ., N} × {1, . . ., N} containing no corner ((x, y), (x + d, y), (x, y + d)), d ̸= 0? Ajtai and Szemerédi [2] proved that r∠(N) = o(N), and various subsequent authors =-=[54, 64]-=- have obtained explicit bounds of the shape r∠(N) ≪ N/(log ∗ N) c . Here log ∗ N is the number of times one must take the logarithm of N in order to produce a number less than 2. 2 This term is one th... |

10 | Recent progress on the Kakeya conjecture
- Katz, Tao
- 2002
(Show Context)
Citation Context ... have proved invaluable, such as the study of the Kakeya and restriction phenomena. We do not touch upon these matters here, referring the reader instead to the article [42] as well as in the surveys =-=[37, 58, 59]-=-. 2. Notation and Basic Definitions Let p be a prime (p will be either 2,3 or 5). Write Fp for the finite field with p elements, which may be identified with Z/pZ, and for an integer n � 1 write F n p... |

9 |
Sur quelques propriétés arithmétiques des presque-périodes
- Bogolyubov
- 1939
(Show Context)
Citation Context ...ns for PFR. Question 10.4. Let A ⊆ F n 2 be a set of density α. Then 2A − 2A contains a subspace with codimension f(α). What is the behaviour of f(α)? Using a Fourier-analytic technique of Bogolyubov =-=[8]-=- one may show that f(α) ≪ α −2 , and a refinement of this technique due to Chang [12] allows one to improve this to f(α) ≪ α −1 log(1/α). We have not been able to rule out the possibility that f(α) ≪ ... |

8 |
Eigenschaften und Abschätzungen von Wirkungsfunktionen. BMwF-GMD-22. Gesellschaft für Mathematik und Datenverarbeitung
- Plünnecke
- 1969
(Show Context)
Citation Context ...log(1/α). We have not been able to rule out the possibility that f(α) ≪ log(1/α), which if true would imply PFR. The proof of Proposition 10.2 uses an important result known as Plünnecke’s inequality =-=[45]-=-, a new proof of which was found by Ruzsa [49]. This states that if A is a subset of any abelian group G, and if |A+A| � K|A|, then we have the inequality |sA−tA| � K s+t |A| for any positive integers... |

7 |
On sets of integers which contain no three elements in arithmetic progression
- Behrend
- 1946
(Show Context)
Citation Context ...th-Brown [36] and Szemerédi [57], and then to r3(N) ≪ N(log log N/ log N) 1/2 by Bourgain [10]. We are still a long way from a complete understanding of r3(N); the best known lower bound is Behrend’s =-=[6]-=- 1946 example showing that r3(N) ≫ Ne −c√ log N . It is natural to define r3(G) for any group G with no 2-torsion (though see [40]). A particularly appealing case, which fits with the discussion of th... |

7 |
Arithmetic progressions in sumsets
- Green
(Show Context)
Citation Context ...s the smallest l for which there is a set A ⊆ {1, . . ., N} of cardinality αN such that A + A does not contain a progression of length l, then Bourgain showed that L(N, 1/10) ≫ exp(c(log N) 1/3 ). In =-=[25]-=- this was improved to L(N, 1/10) ≫ exp(c(log N) 1/2 ). An example of Ruzsa [50] shows that L(N, 1/10) ≪ exp(cǫ(log N) 2/3+ǫ ). It seems as though the natural finite field analogue of Problem 9.1 invol... |

5 |
On a question of Gowers, Ann
- Vu
(Show Context)
Citation Context ... of the largest subset of {1, . . ., N} × {1, . . ., N} containing no corner ((x, y), (x + d, y), (x, y + d)), d ̸= 0? Ajtai and Szemerédi [2] proved that r∠(N) = o(N), and various subsequent authors =-=[54, 64]-=- have obtained explicit bounds of the shape r∠(N) ≪ N/(log ∗ N) c . Here log ∗ N is the number of times one must take the logarithm of N in order to produce a number less than 2. 2 This term is one th... |

4 |
On a problem of Gowers, preprint
- Shkredov
(Show Context)
Citation Context ...f graph regularity in the excellent survey [39]: now the more descriptive term “counting lemma” is popular (cf. [23, 26, 43]).FINITE FIELD MODELS IN ARITHMETIC COMBINATORICS 9 Very recently Shkredov =-=[53]-=- produced the first “sensible” bound r∠(N) ≪ N/(log log log N) c . In this section we give the finite field version of his argument, in which the details are greatly simplified. Let G be an abelian gr... |

4 |
sets containing no arithmetic progressions
- Integer
- 1990
(Show Context)
Citation Context ...ssed by Klaus Roth [46]. Define r3(N) to be the answer to Problem 1.1. Roth proved that r3(N) ≪ N/ log log N, a bound which was improved to N(log N) −c independently by Heath-Brown [36] and Szemerédi =-=[57]-=-, and then to r3(N) ≪ N(log log N/ log N) 1/2 by Bourgain [10]. We are still a long way from a complete understanding of r3(N); the best known lower bound is Behrend’s [6] 1946 example showing that r3... |

3 |
Progression-free sets in finite abelian groups
- Lev
(Show Context)
Citation Context ...mplete understanding of r3(N); the best known lower bound is Behrend’s [6] 1946 example showing that r3(N) ≫ Ne −c√ log N . It is natural to define r3(G) for any group G with no 2-torsion (though see =-=[40]-=-). A particularly appealing case, which fits with the discussion of this article, is G = Fn 3 . In this case it turns out that the four proofs [10, 36, 46, 57] can all be adapted to give the following... |

3 |
rotating needles to stability of waves: emerging connections between combinatorics, analysis
- From
(Show Context)
Citation Context ... have proved invaluable, such as the study of the Kakeya and restriction phenomena. We do not touch upon these matters here, referring the reader instead to the article [42] as well as in the surveys =-=[37, 58, 59]-=-. 2. Notation and Basic Definitions Let p be a prime (p will be either 2,3 or 5). Write Fp for the finite field with p elements, which may be identified with Z/pZ, and for an integer n � 1 write F n p... |

1 |
Arithmetic progressions in sumsets, in A Tribute to Paul Erdős, CUP
- Bourgain
- 1990
(Show Context)
Citation Context ...what more miscellaneous nature. This section concerns the following problem. Problem 9.1. Let A ⊆ {1, . . ., N} be a set of size N/10 (say). Must A + A contain a long arithmetic progression? Bourgain =-=[9]-=- proved that the answer is “yes”; A + A must contain a surprisingly long arithmetic progression. If L(N, α) is the smallest l for which there is a set A ⊆ {1, . . ., N} of cardinality αN such that A +... |

1 |
Extensions of generalized product caps, Designs, Codes and Cryptography 31 (2004), 5–14. Available at http://www.mathi.uni-heidelberg.de/˜yves/Papers/ExtProd.pdf
- Edel
(Show Context)
Citation Context ...exity in Rn ). The best known lower bounds on r3(Fn 3 ) come from design theory, where a set in Fn 3 with no 3-term AP is known as a cap. Write f(n) for the cardinality of the largest cap in Fn 3. In =-=[15]-=- one finds the estimate log µ(3) := lim sup 3(f(n)) � 0.724851, n→∞ n which seems to be the best known. In that paper it is stated as an interesting research problem to determine if µ(3) = 1. I believ... |

1 |
lecture notes on Freiman’s theorem, notes. Available at http://www.dpmms.cam.ac.uk/˜bjg23
- Edinburgh
(Show Context)
Citation Context ...ontext of general abelian groups G. However, the issues are of a rather different nature to those discussed in §8. Freiman’s original work concerned subsets of Z, and was quite geometric in feel. See =-=[7, 17, 27]-=- for a further discussion. Ruzsa’s proof [48] has proved much more adaptable, and recently Ruzsa and the author [32] were able to obtain a structure theorem for sets with small doubling which is valid... |

1 |
with small sumset and rectification
- Sets
(Show Context)
Citation Context ...ets of subspaces, and large subsets of them. In fact, these are the only such examples as was shown by Imre Ruzsa [47]. The best known bounds for a result of this type are due to Ruzsa and the author =-=[31]-=-: Theorem 10.1 (Freiman’s theorem in F∞ 2 ). Let A ⊆ F∞2 be a finite set with |A + A| � K|A|. Then A is contained within a coset of some subgroup H � F∞ 2 with |H| � K222K2 −2 |A|. A version of this r... |

1 |
Pseudo-random graphs, survey article. Available at http://www.math.princeton.edu/˜bsudakov
- Krivelevich, Sudakov
(Show Context)
Citation Context ... [13] later defined quasi-randomness for subsets of Z/NZ. Quasirandomness has been most thoroughly explored in the context of graphs, for which the reader should consult the excellent survey articles =-=[38, 39]-=-. The notions of uniformity in Z/NZ and in Fn p differ in little more than notation. As an example of uniformity/quasirandomness at work, and to get comfortable with the notation, let us prove that un... |

1 |
The counting lemma for k-uniform hypergraphs, submitted
- Nagle, Rödl, et al.
(Show Context)
Citation Context ... example [33, §5]. The phrase “key lemma” was used for a related concept in the theory of graph regularity in the excellent survey [39]: now the more descriptive term “counting lemma” is popular (cf. =-=[23, 26, 43]-=-).FINITE FIELD MODELS IN ARITHMETIC COMBINATORICS 9 Very recently Shkredov [53] produced the first “sensible” bound r∠(N) ≪ N/(log log log N) c . In this section we give the finite field version of h... |

1 |
Recent progress on the restriction phenomenon
- Tao
- 2004
(Show Context)
Citation Context ... have proved invaluable, such as the study of the Kakeya and restriction phenomena. We do not touch upon these matters here, referring the reader instead to the article [42] as well as in the surveys =-=[37, 58, 59]-=-. 2. Notation and Basic Definitions Let p be a prime (p will be either 2,3 or 5). Write Fp for the finite field with p elements, which may be identified with Z/pZ, and for an integer n � 1 write F n p... |