For every finite set of real numbers, the pairwise sums or products form a bigger set
In
arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every
finite set of
integers, at least one of , the set of pairwise sums or , the set of pairwise products form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants c and such that for any non-empty set
The set of pairwise sums is and is called
sum set of .
The set of pairwise products is and is called the product set of .
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 and Szemerédi over the integers,[1] but is thought to hold over the real numbers.
Conjecture: For any set one has
The asymptotic parameter in the o(1) notation is |A|.
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.[3] The best lower bound on for this set is due to
Kevin Ford.[4]
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, for example an
arithmetic progression has the lower bound on the product set . Xu and Zhou [6] proved that for any dense subset of an arithmetic progression in integers, which is sharp up to the factor in the exponent.
Conversely, the set satisfies , but has many sums: . 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 sum set have cardinality ; that is, with high probability the sum set generates no repeated elements, and the same for the product set.
Sharpness of the conjecture
Erdős and
Szemerédi give an example of a sufficiently
smooth set of integers with the bound:
The following table summarises 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]
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.[19] Konyagin and Rudnev[20] 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[21] 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) [22]): Let be prime and let with for some . Then one has 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) [22]): Let be a subset of a finite field so that for some and suppose that . Then there exists a sub-field with , 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 with respect to the
characteristic of and when is 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.
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[29] 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:[30]
Theorem (Garaev (2007) ): Let . Then .
This is optimal in the range .
This result was extended to finite fields of non-prime order by Vinh[31] in 2011.
Variants and generalisations
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)[32]): 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.