Erdős–Szemerédi Theorem

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 ,

:$\max( , A+A, , , A \cdot A, ) \geq c , A, ^$.
It was proved by Paul Erdős and Endre Szemerédi in 1983.. 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.

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. More precisely, it proposes that, for any set , one has

$\max(, A+A, ,, AA, ) \geq , A, ^.$

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. The best lower bound on for this set is due to Kevin Ford.

This example is an instance of the ''Few Sums, Many Products'' 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 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: $, B+B, = \binom$. 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 $\binom$; 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

$\max\ \leq , A, ^ e^$.

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 .

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.

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. Konyagin and Rudnev 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 when is a (large) prime, for every subset , the inequality holds for an absolute constant . This conjecture had also been formulated in the 1990s by Wigderson, 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)): 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.

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 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:

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

This is optimal in the range .

This result was extended to finite fields of non-prime order by Vinh 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)): 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 show that
* : Murphy, Roche-Newton and Shkredov 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

References

External links

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{{DEFAULTSORT:Erdos-Szemeredi theorem
Additive combinatorics
Sumsets
Theorems in discrete mathematics
Theorems in number theory