In additive number theory and combinatorics, a restricted sumset has the form where
are finite nonempty subsets of a field F and
is a constant non-zero function, for example
is the usual sumset
which is denoted by
{\displaystyle nA}
When S is written as
which is denoted by
Note that |S| > 0 if and only if there exist
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
, i.e. we're using modular arithmetic.
It can be generalised to arbitrary (not necessarily abelian) groups using a Dyson transform.
are subsets of a group
is the size of the smallest nontrivial subgroup of
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
, 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
, every element of
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] 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.
A. Dias da Silva and Y. O. Hamidoune in 1994[10] who showed that 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] A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz.
be a polynomial over a field
Suppose that the coefficient of the monomial
is the total degree of
are finite subsets of
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.