Restricted sumset

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

${\displaystyle S=\{a_{1}+\cdots +a_{n}:\ a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}\ \mathrm {and} \ P(a_{1},\ldots ,a_{n})\not =0\},}${\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 ${\displaystyle A_{1},\ldots ,A_{n}}${\displaystyle A_{1},\ldots ,A_{n}} are finite nonempty subsets of a field F and ${\displaystyle P(x_{1},\ldots ,x_{n})}${\displaystyle P(x_{1},\ldots ,x_{n})} is a polynomial over F.

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

When

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

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

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

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 ${\displaystyle \mathbb {Z} /p\mathbb {Z} }${\displaystyle \mathbb {Z} /p\mathbb {Z} } we have the inequality ^{[1] [2] [3]}

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

where ${\displaystyle A+B:=\{a+b{\pmod {p}}\mid a\in A,b\in B\}}${\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 ${\displaystyle A,B}${\displaystyle A,B} are subsets of a group ${\displaystyle G}${\displaystyle G}, then^[4]

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

where ${\displaystyle p(G)}${\displaystyle p(G)} is the size of the smallest nontrivial subgroup of ${\displaystyle G}${\displaystyle G} (we set it to ${\displaystyle 1}${\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 ${\displaystyle \mathbb {Z} /n\mathbb {Z} }${\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 ${\displaystyle \mathbb {Z} /p\mathbb {Z} }${\displaystyle \mathbb {Z} /p\mathbb {Z} }, every element of ${\displaystyle \mathbb {Z} /p\mathbb {Z} }${\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]

The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that ${\displaystyle |2^{\wedge }A|\geq \min\{p,,2円|A|-3\}}${\displaystyle |2^{\wedge }A|\geq \min\{p,,2円|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

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

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 ${\displaystyle f(x_{1},\ldots ,x_{n})}${\displaystyle f(x_{1},\ldots ,x_{n})} be a polynomial over a field ${\displaystyle F}${\displaystyle F}. Suppose that the coefficient of the monomial ${\displaystyle x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}}${\displaystyle x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}} in ${\displaystyle f(x_{1},\ldots ,x_{n})}${\displaystyle f(x_{1},\ldots ,x_{n})} is nonzero and ${\displaystyle k_{1}+\cdots +k_{n}}${\displaystyle k_{1}+\cdots +k_{n}} is the total degree of ${\displaystyle f(x_{1},\ldots ,x_{n})}${\displaystyle f(x_{1},\ldots ,x_{n})}. If ${\displaystyle A_{1},\ldots ,A_{n}}${\displaystyle A_{1},\ldots ,A_{n}} are finite subsets of ${\displaystyle F}${\displaystyle F} with ${\displaystyle |A_{i}|>k_{i}}${\displaystyle |A_{i}|>k_{i}} for ${\displaystyle i=1,\ldots ,n}${\displaystyle i=1,\ldots ,n}, then there are ${\displaystyle a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}}${\displaystyle a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}} such that ${\displaystyle f(a_{1},\ldots ,a_{n})\neq 0}${\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]

• Polynomial method in combinatorics

• Geroldinger, Alfred; Ruzsa, Imre Z., eds. (2009). Combinatorial number theory and additive group theory. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. ISBN 978-3-7643-8961-1. Zbl 1177.11005.
• Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003.

• Weisstein, Eric W. "Erdős-Heilbronn Conjecture". MathWorld .

• Augustin-Louis Cauchy
• Sumsets
• Additive combinatorics
• Additive number theory

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