Restricted sumset

In additive number theory and combinatorics, a restricted sumset has the form

${\displaystyle S=\{a_{1}+\cdots +a_{n}:\ a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}\ \mathrm {and} \ P(a_{1},\ldots ,a_{n})\not =0\},}$

where ${\displaystyle A_{1},\ldots ,A_{n}}$ are finite nonempty subsets of a field F and ${\displaystyle P(x_{1},\ldots ,x_{n})}$ is a polynomial over F.

If ${\displaystyle P}$ is a constant non-zero function, for example ${\displaystyle P(x_{1},\ldots ,x_{n})=1}$ for any ${\displaystyle x_{1},\ldots ,x_{n}}$, then ${\displaystyle S}$ is the usual sumset ${\displaystyle A_{1}+\cdots +A_{n}}$ which is denoted by ${\displaystyle nA}$ if ${\displaystyle A_{1}=\cdots =A_{n}=A.}$

When

${\displaystyle P(x_{1},\ldots ,x_{n})=\prod _{1\leq i<j\leq n}(x_{j}-x_{i}),}$

S is written as ${\displaystyle A_{1}\dotplus \cdots \dotplus A_{n}}$ which is denoted by ${\displaystyle n^{\wedge }A}$ if ${\displaystyle A_{1}=\cdots =A_{n}=A.}$

Note that |S| > 0 if and only if there exist ${\displaystyle a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}}$ with ${\displaystyle P(a_{1},\ldots ,a_{n})\not =0.}$

Cauchy–Davenport theorem

The Cauchy–Davenport theorem, named after Augustin Louis Cauchy and Harold Davenport, asserts that for any prime p and nonempty subsets A and B of the prime order cyclic group ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ we have the inequality^[1][2][3]

${\displaystyle |A+B|\geq \min\{p,\,|A|+|B|-1\}}$

where ${\displaystyle A+B:=\{a+b{\pmod {p}}\mid a\in A,b\in B\}}$, i.e. we're using modular arithmetic. It can be generalised to arbitrary (not necessarily abelian) groups using a Dyson transform. If ${\displaystyle A,B}$ are subsets of a group ${\displaystyle G}$, then^[4]

${\displaystyle |A+B|\geq \min\{p(G),\,|A|+|B|-1\}}$

where ${\displaystyle p(G)}$ is the size of the smallest nontrivial subgroup of ${\displaystyle G}$ (we set it to ${\displaystyle 1}$ if there is no such subgroup).

We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2n−1 elements in the cyclic group ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$, there are n elements that sum to zero modulo n. (Here n does not need to be prime.)^[5][6]

A direct consequence of the Cauchy-Davenport theorem is: Given any sequence S of p−1 or more nonzero elements, not necessarily distinct, of ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$, every element of ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ can be written as the sum of the elements of some subsequence (possibly empty) of S.^[7]

Kneser's theorem generalises this to general abelian groups.^[8]

Erdős–Heilbronn conjecture

The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that ${\displaystyle |2^{\wedge }A|\geq \min\{p,\,2|A|-3\}}$ if p is a prime and A is a nonempty subset of the field Z/pZ.^[9] This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994^[10] who showed that

${\displaystyle |n^{\wedge }A|\geq \min\{p(F),\ n|A|-n^{2}+1\},}$

where A is a finite nonempty subset of a field F, and p(F) is a prime p if F is of characteristic p, and p(F) = ∞ if F is of characteristic 0. Various extensions of this result were given by Noga Alon, M. B. Nathanson and I. Ruzsa in 1996,^[11] Q. H. Hou and Zhi-Wei Sun in 2002,^[12] and G. Karolyi in 2004.^[13]

Combinatorial Nullstellensatz

A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz.^[14] Let ${\displaystyle f(x_{1},\ldots ,x_{n})}$ be a polynomial over a field ${\displaystyle F}$. Suppose that the coefficient of the monomial ${\displaystyle x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}}$ in ${\displaystyle f(x_{1},\ldots ,x_{n})}$ is nonzero and ${\displaystyle k_{1}+\cdots +k_{n}}$ is the total degree of ${\displaystyle f(x_{1},\ldots ,x_{n})}$. If ${\displaystyle A_{1},\ldots ,A_{n}}$ are finite subsets of ${\displaystyle F}$ with ${\displaystyle |A_{i}|>k_{i}}$ for ${\displaystyle i=1,\ldots ,n}$, then there are ${\displaystyle a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}}$ such that ${\displaystyle f(a_{1},\ldots ,a_{n})\neq 0}$.

This tool was rooted in a paper of N. Alon and M. Tarsi in 1989,^[15] and developed by Alon, Nathanson and Ruzsa in 1995–1996,^[11] and reformulated by Alon in 1999.^[14]

See also

• Polynomial method in combinatorics