Restricted sumset

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

$S=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n \ \mathrm{and}\ P(a_1,\ldots,a_n)\not=0\},$

where $A_1,\ldots,A_n$ are finite nonempty subsets of a field $F$ and $P(x_1,\ldots,x_n)$ is a polynomial over $F$.

When $P(x_1,\ldots,x_n)=1$, $S$ is the usual sumset $A_1+\cdots+A_n$ which is denoted by $nA$ if $A_1=\cdots=A_n=A$; when

$P(x_1,\ldots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i),$

$S$ is written as $A_1\dotplus\cdots\dotplus A_n$ which is denoted by $n^{\wedge} A$ if $A_1=\cdots=A_n=A$. Note that $|S|>0$ if and only if there exist $a_1\in A_1,\ldots,a_n\in A_n$ with $P(a_1,\ldots,a_n)\not=0$.

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 field $\Bbb Z/p\Bbb Z$ we have the inequality

$|A+B|\ge\min\{p,\ |A|+|B|-1\}.\,$

Erdős–Heilbronn conjecture

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

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

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 $p(F)=\infty$ if $F$ is of characteristic 0. Various extensions of this result were given by Noga Alon, M. B. Nathanson and I. Ruzsa in 1996, Q. H. Hou and Zhi-Wei Sun in 2002, and G. Karolyi in 2004.

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. Let $f(x_1,\ldots,x_n)$ be a polynomial over a field $F$. Suppose that the coefficient of the monomial $x_1^{k_1}\cdots x_n^{k_n}$ in $f(x_1,\ldots,x_n)$ is nonzero and $k_1+\cdots+k_n$ is the total degree of $f(x_1,\ldots,x_n)$. If $A_1,\ldots,A_n$ are finite subsets of $F$ with $|A_i|>k_i$ for $i=1,\ldots,n$, then there are $a_1\in A_1,\ldots,a_n\in A_n$ such that $f(a_1,\ldots,a_n)\not=0$.

The method using the combinatorial Nullstellensatz is also called the polynomial method. This tool was rooted in a paper of N. Alon and M. Tarsi in 1989, and developed by Alon, Nathanson and Ruzsa in 1995-1996, and reformulated by Alon in 1999.

External links

• Sun, Zhi-Wei (2006). "An additive theorem and restricted sumsets". Math. Res. Lett. , no. 15 (6): 1263–1276. arXiv:math.CO/0610981. 
• Zhi-Wei Sun: On some conjectures of Erdős-Heilbronn, Lev and Snevily (PDF), a survey talk.
• Weisstein, Eric W., "Erdos-Heilbronn Conjecture" from MathWorld.