Erdős–Szemerédi theorem explained

In arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every finite set of integers, at least one of the sets and (the sets of pairwise sums and pairwise products, respectively) form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants and such that, for any non-empty set,

max ( |A + A|, |A ⋅ A| ) \geq c |A| ^{1+\varepsilon} .It was proved by Paul Erdős and Endre Szemerédi in 1983.^[1] The notation denotes the cardinality of the set .

The set of pairwise sums is and is called the sumset of .

The set of pairwise products is and is called the product set of ; it is also written .

The theorem is a version of the maxim that additive structure and multiplicative structure cannot coexist. It can also be viewed as an assertion that the real line does not contain any set resembling a finite subring or finite subfield; it is the first example of what is now known as the sum-product phenomenon, which is now known to hold in a wide variety of rings and fields, including finite fields.^[2]

Sum-Product Conjecture

The sum-product conjecture informally says that one of the sum set or the product set of any set must be nearly as large as possible. It was originally conjectured by Erdős in 1974 to hold whether is a set of integers, reals, or complex numbers.^[3] More precisely, it proposes that, for any set, one has

max (|A + A|,|AA|) \geq |A| ^{2-o(1)} .

The asymptotic parameter in the notation is .

Examples

If, then using asymptotic notation, with the asymptotic parameter. Informally, this says that the sum set of does not grow. On the other hand, the product set of satisfies a bound of the form for all . This is related to the Erdős multiplication table problem.^[4] The best lower bound on for this set is due to Kevin Ford.

This example is an instance of the Few Sums, Many Products^[5] version of the sum-product problem of György Elekes and Imre Z. Ruzsa. A consequence of their result is that any set with small additive doubling (such as an arithmetic progression) has the lower bound on the product set . Xu and Zhou proved^[6] that for any dense subset of an arithmetic progression in integers, which is sharp up to the in the exponent.

Conversely, the set satisfies, but has many sums:

|B + B| = \binom{|B| +1 }{ 2 } . This bound comes from considering the binary representation of a number. The set is an example of a geometric progression.

For a random set of numbers, both the product set and the sumset have cardinality

\binom{n +1 }{ 2 } ; that is, with high probability, neither the sumset nor the product set generates repeated elements.

Sharpness of the conjecture

Erdős and Szemerédi give an example of a sufficiently smooth set of integers with the bound

max \{|A + A|,|A ⋅ A|\} \leq |A| ^{2} e ^{- (c log |A|)/(loglog |A|)} .

This shows that the term in the conjecture is necessary.

Extremal cases

Often studied are the extreme cases of the hypothesis:

• few sums, many product (FSMP): if, then, and
• few products, many sums (FPMS): if, then .^[7]

History and current results

The following table summarizes progress on the sum-product problem over the reals. The exponents 1/4 of György Elekes and 1/3 of József Solymosi are considered milestone results within the citing literature. All improvements after 2009 are of the form, and represent refinements of the arguments of Konyagin and Shkredov.^[8]

Exponent where for !Year!Exponent!Author(s)!Comments

1983	non-explicit	Erdős and Szemerédi	Only for and of the form instead of .
1997		Nathanson^[9]	Only for and of the form instead of .
1998		Ford ^[10]	Only for and of the form instead of .
1997		Elekes ^[11]	Of the form . Valid also over
2005		Solymosi^[12]	Valid also over
2009		Solymosi ^[13]
2015		Konyagin and Shkredov
2016		Konyagin and Shkredov ^[14]
2016		Rudnev, Shkredov and Stevens ^[15]
2019		Shakan ^[16]
2020		Rudnev and Stevens ^[17]	Current record

Complex numbers

Proof techniques involving only the Szemerédi–Trotter theorem extend automatically to the complex numbers, since the Szemerédi-Trotter theorem holds over by a theorem of Tóth.^[18] Konyagin and Rudnev^[19] matched the exponent of 4/3 over the complex numbers. The results with exponents of the form have not been matched over the complex numbers.

Over finite fields

The sum-product problem is particularly well-studied over finite fields. Motivated by the finite field Kakeya conjecture, Wolff conjectured that for every subset, where is a (large) prime, that for an absolute constant . This conjecture had also been formulated in the 1990s by Wigderson^[20] motivated by randomness extractors.

Note that the sum-product problem cannot hold in finite fields unconditionally due to the following example:

Example: Let be a finite field and take . Then since is closed under addition and multiplication,, and so . This pathological example extends to taking to be any sub-field of the field in question.

Qualitatively, the sum-product problem has been solved over finite fields:

Theorem (Bourgain, Katz, Tao (2004)):^[21] Let be prime and let with for some . Then for some .

Bourgain, Katz, and Tao extended this theorem to arbitrary fields. Informally, the following theorem says that if a sufficiently large set does not grow under either addition or multiplication, then it is mostly contained in a dilate of a sub-field.

Theorem (Bourgain, Katz, Tao (2004)): Let be a subset of a finite field so that for some, and suppose that. Then there exists a sub-field with, an element, and a set with so that .

They suggest that the constant may be independent of .

Quantitative results towards the finite field sum-product problem in typically fall into two categories: when is small or large with respect to the characteristic of . This is because different types of techniques are used in each setting.

Small sets

In this regime, let be a field of characteristic . Note that the field is not always finite. When this is the case, and the characteristic of is zero, then the -constraint is omitted.

Exponent where for !Year!Exponent!-constraint: !Author(s)!Comments

2004	unquantified		Bourgain, Katz, Tao	For finite only.
2007			Garaev^[22]	For finite only. The -constraint involves a factor of
2008			Katz, Shen	For finite only.
2009			Bourgain, Garaev^[23]	For finite only. removed by Li.^[24]
2012			Rudnev^[25]	For finite only.
2016			Roche-Newton, Rudnev, Shkredov^[26]
2016			Rudnev, Shkredov, Shakan	This result is the best of three contemporaneous results.
2021			Mohammadi, Stevens ^[27]	Current record. Extends to difference sets and ratio sets.

In fields with non-prime order, the -constraint on can be replaced with the assumption that does not have too large an intersection with any subfield. The best work in this direction is due to Li and Roche-Newton^[28] attaining an exponent of in the notation of the above table.

Large sets

When for prime, the sum-product problem is considered resolved due to the following result of Garaev:^[29]

Theorem (Garaev (2007)): Let . Then .

This is optimal in the range .

This result was extended to finite fields of non-prime order by Vinh^[30] in 2011.

Variants and generalizations

Other combinations of operators

Bourgain and Chang proved unconditional growth for sets, as long as one considers enough sums or products:

Theorem (Bourgain, Chang (2003)):^[31] Let . Then there exists so that for all, one has .

In many works, addition and multiplication are combined in one expression. With the motto addition and multiplication cannot coexist, one expects that any non-trivial combination of addition and multiplication of a set should guarantee growth. Note that in finite settings, or in fields with non-trivial subfields, such a statement requires further constraints.

Sets of interest include (results for):

• : Stevens and Warren^[32] show that
• : Murphy, Roche-Newton and Shkredov^[33] show that
• : Stevens and Warren show that
• : Stevens and Rudnev show that

See also

• Sum-free set
• Restricted sumset
• Additive combinatorics
• Additive number theory
• Multiplicative number theory

External links

• How a Strange Grid Reveals Hidden Connections Between Simple Numbers . . Kevin . Hartnett . 6 February 2019.

Notes and References

1. .
2. .
3. P. Erdős: Some recent problems and results in graph theory, combinatorics and number theory, Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congress. Numer. XVII, pp. 3--14, Utilitas Math., Winnipeg, Man., 1976 MR54 #10023; Zentralblatt 352.05024.
4. Paul. Erdős. Paul Erdős. An asymptotic inequality in the theory of numbers. Vestnik Leningrad. Univ.. 15. 1960. 41–49. 0126424.
5. Elekes Gy.. György. Ruzsa. Imre Z.. 2003-08-01. Few sums, many products. Studia Scientiarum Mathematicarum Hungarica. 40. 3. 301–308. 10.1556/sscmath.40.2003.3.4. 0081-6906.
6. 2201.00104 . math.NT . Max Wenqiang . Xu . Yunkun . Zhou . On product sets of arithmetic progressions . 2022.
7. Murphy. Brendan. Rudnev. Misha. Shkredov. Ilya. Shteinikov. Yuri. 2019. On the few products, many sums problem. Journal de Théorie des Nombres de Bordeaux. 31. 3. 573–602. 10.5802/jtnb.1095. 1712.00410. 119665080.
8. Konyagin. S. V.. Shkredov. I. D.. August 2015. On sum sets of sets having small product set. Proceedings of the Steklov Institute of Mathematics. 290. 1. 288–299. 10.1134/s0081543815060255. 0081-5438. 1503.05771. 117359454.
9. Nathanson. Melvyn B.. 1997. On sums and products of integers. Proceedings of the American Mathematical Society. 125. 1. 9–16. 10.1090/s0002-9939-97-03510-7. free. 0002-9939.
10. Ford. Kevin. 1998. Sums and products from a finite set of real numbers. The Ramanujan Journal. 2. 1/2. 59–66. 10.1023/a:1009709908223. 195302784. 1382-4090.
11. Elekes. György. 1997. On the number of sums and products. Acta Arithmetica. 81. 4. 365–367. 10.4064/aa-81-4-365-367. 0065-1036. free.
12. Solymosi. József. August 2005. On the number of sums and products. Bulletin of the London Mathematical Society. 37. 4. 491–494. 10.1112/s0024609305004261. 56432429 . 0024-6093.
13. Solymosi. József. October 2009. Bounding multiplicative energy by the sumset. Advances in Mathematics. 222. 2. 402–408. 10.1016/j.aim.2009.04.006. free. 0001-8708. 0806.1040.
14. Konyagin. S. V.. Shkredov. I. D.. August 2016. New results on sums and products in ℝ. Proceedings of the Steklov Institute of Mathematics. 294. 1. 78–88. 10.1134/s0081543816060055. 126099880. 0081-5438.
15. Rudnev. Misha. Shkredov. Ilya. Stevens. Sophie. 2019-09-10. On the energy variant of the sum-product conjecture. Revista Matemática Iberoamericana. 36. 1. 207–232. 10.4171/rmi/1126. 0213-2230. 1607.05053. 119122310.
16. Shakan. George. 2018-07-03. On higher energy decompositions and the sum–product phenomenon. Mathematical Proceedings of the Cambridge Philosophical Society. 167. 3. 599–617. 10.1017/s0305004118000506. 0305-0041. 1803.04637. 119693920.
17. 2005.11145. Misha . Rudnev. Sophie . Stevens. An update on the sum-product problem. 2022. Mathematical Proceedings of the Cambridge Philosophical Society. 173. 2. 411–430. 10.1017/S0305004121000633.
18. Tóth. Csaba D.. February 2015. The Szemerédi-Trotter theorem in the complex plane. Combinatorica. 35. 1. 95–126. 10.1007/s00493-014-2686-2. 0209-9683. math/0305283. 13237229.
19. Konyagin. Sergei V.. Rudnev. Misha. January 2013. On New Sum-Product--Type Estimates. SIAM Journal on Discrete Mathematics. 27. 2. 973–990. 10.1137/120886418. 0895-4801. 1111.4977. 207065775.
20. Trevisan. Luca. 2009-06-20. Guest column. ACM SIGACT News. 40. 2. 50–66. 10.1145/1556154.1556170. 12566158. 0163-5700.
21. Bourgain . Jean . Katz . Nets . Tao . Terence . 2004-02-01 . A sum-product estimate in finite fields, and applications . Geometric and Functional Analysis . 14 . 1 . 27–57 . math/0301343 . 10.1007/s00039-004-0451-1 . 1016-443X . 14097626.
22. Garaev. M. Z.. 2010-07-08. An Explicit Sum-Product Estimate in Fp. International Mathematics Research Notices. 10.1093/imrn/rnm035. 1073-7928. math/0702780.
23. Bourgain . J. . Jean Bourgain . Garaev . M. Z. . On a variant of sum-product estimates and explicit exponential sum bounds in prime fields. Mathematical Proceedings of the Cambridge Philosophical Society. 2008. 146. 1. 1. 10.1017/S0305004108001230. 2008MPCPS.146....1B. 120185078.
24. Li. Liangpan. 2011. Slightly improved sum-product estimates in fields of prime order. Acta Arithmetica. 147. 2. 153–160. 10.4064/aa147-2-4. 0065-1036. 0907.2051. 15954935.
25. Rudnev. Misha. 2011-08-25. An Improved Sum–Product Inequality in Fields of Prime Order. International Mathematics Research Notices. 2012. 16. 3693–3705. 10.1093/imrn/rnr158. 1687-0247. 1011.2738.
26. Roche-Newton. Oliver. Rudnev. Misha. Shkredov. Ilya D.. New sum-product type estimates over finite fields. Advances in Mathematics. 2016. 293. 589–605 . 10.1016/j.aim.2016.02.019 . free. 1408.0542.
27. Mohammadi . Ali. Stevens . Sophie. Attaining the exponent 5/4 for the sum-product problem in finite fields. 2023. 2103.08252. International Mathematics Research Notices. 4. 2023. 3516–3532. 10.1093/imrn/rnab338.
28. Li. Liangpan. Roche-Newton. Oliver. January 2011. An improved sum-product estimate for general finite fields. SIAM Journal on Discrete Mathematics. 25. 3. 1285–1296. 10.1137/110823122. 0895-4801. 1101.5348. 7024012.
29. Garaev. M. Z.. 2008-04-14. The sum-product estimate for large subsets of prime fields. Proceedings of the American Mathematical Society. 136. 8. 2735–2739. 10.1090/s0002-9939-08-09386-6. 0002-9939. 0706.0702. 16064726.
30. Vinh. Le Anh. November 2011. The Szemerédi–Trotter type theorem and the sum-product estimate in finite fields. European Journal of Combinatorics. 32. 8. 1177–1181. 10.1016/j.ejc.2011.06.008. free. 0195-6698. 0711.4427.
31. Bourgain . Jean . Chang . Mei-Chu . 2003-11-25 . On the size of k -fold sum and product sets of integers . Journal of the American Mathematical Society . 17 . 2 . 473–497 . math/0309055 . 2003math......9055B . 10.1090/s0894-0347-03-00446-6 . 0894-0347 . 15154515.
32. Stevens . Sophie. Warren . Audie. On sum sets of convex functions. 2022. 2102.05446. Electronic Journal of Combinatorics. 29. 2. P2.18. 10.37236/10852 . free.
33. Murphy. Brendan. Roche-Newton. Oliver. Shkredov. Ilya D.. January 2017. Variations on the Sum-Product Problem II. SIAM Journal on Discrete Mathematics. en. 31. 3. 1878–1894. 10.1137/17M112316X. 0895-4801. 1703.09549. 207074281.