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Cauchy–Davenport theorem : ウィキペディア英語版
Restricted sumset
In additive number theory and combinatorics, a restricted sumset has the form
:S=\,
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_(x_j-x_i),
''S'' is written as A_1\dotplus\cdots\dotplus A_n which is denoted by n^ 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 prime order cyclic group Z/''p''Z we have the inequality〔Nathanson (1996) p.44〕〔Geroldinger & Ruzsa (2009) pp.141–142〕
:|A+B|\ge\min\.\,
We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2''n''−1 elements in Z/''n'', there are ''n'' elements that sums to zero modulo ''n''. (Here ''n'' does not need to be prime.)〔Nathanson (1996) p.48〕〔Geroldinger & Ruzsa (2009) p.53〕
A direct consequence of the Cauchy-Davenport theorem is: Given any set ''S'' of ''p''−1 or more elements, not necessarily distinct, of Z/''p''Z, every element of Z/''p''Z can be written as the sum of the elements of some subset (possibly empty) of ''S''.〔Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.〕
Kneser's theorem generalises this to finite abelian groups.〔Geroldinger & Ruzsa (2009) p.143〕

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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