Restricted sumset


In additive number theory and combinatorics, a restricted sumset has the form
where are finite nonempty subsets of a field F and is a polynomial over F.
When, S is the usual sumset which is denoted by nA if ; when
S is written as which is denoted by if. Note that |S| > 0 if and only if there exist with.

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 Z/pZ we have the inequality
We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2n−1 elements in Z/n, there are n elements that sums to zero modulo n.
A direct consequence of the Cauchy-Davenport theorem is: Given any set S of p−1 or more nonzero elements, not necessarily distinct, of Z/pZ, every element of Z/pZ can be written as the sum of the elements of some subset of S.
Kneser's theorem generalises this to general abelian groups.

Erdős–Heilbronn conjecture

The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that if p is a prime and A is a nonempty subset of the field Z/pZ. This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994
who showed that
where A is a finite nonempty subset of a field F, and p is a prime p if F is of characteristic p, and p = ∞ 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 be a polynomial over a field F. Suppose that the coefficient of the monomial in is nonzero and is the total degree of. If are finite subsets of F with for, then there are such that.
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.