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| In [[additive number theory]] and [[combinatorics]], a '''restricted sumset''' has the form
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| :<math>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\},</math>
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| where <math> A_1,\ldots,A_n</math> are finite nonempty subsets of a [[field (mathematics)|field]] ''F'' and <math>P(x_1,\ldots,x_n)</math> is a polynomial over ''F''.
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| When <math>P(x_1,\ldots,x_n)=1</math>, ''S'' is the usual [[sumset]] <math>A_1+\cdots+A_n</math> which is denoted by ''nA'' if <math>A_1=\cdots=A_n=A</math>; when
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| :<math>P(x_1,\ldots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i),</math>
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| ''S'' is written as <math>A_1\dotplus\cdots\dotplus A_n</math> which is denoted by <math>n^{\wedge} A</math> if <math>A_1=\cdots=A_n=A</math>. Note that |''S''| > 0 if and only if there exist <math>a_1\in A_1,\ldots,a_n\in A_n</math> with <math>P(a_1,\ldots,a_n)\not=0</math>.
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| == Cauchy–Davenport theorem ==
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| 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<ref>Nathanson (1996) p.44</ref><ref name=GR1412>Geroldinger & Ruzsa (2009) pp.141–142</ref>
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| :<math>|A+B|\ge\min\{p,\ |A|+|B|-1\}.\,</math>
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| We may use this to deduce the [[Erdős-Ginzburg-Ziv theorem]]: given any 2''n''−1 elements of '''Z'''/''n'', there is a non-trivial subset that sums to zero modulo ''n''. (Here ''n'' does not need to be prime.)<ref>Nathanson (1996) p.48</ref><ref name=GR53>Geroldinger & Ruzsa (2009) p.53</ref>
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| 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''.<ref>Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.</ref>
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| [[Kneser's theorem (combinatorics)|Kneser's theorem]] generalises this to finite abelian groups.<ref name=GR143>Geroldinger & Ruzsa (2009) p.143</ref>
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| == Erdős–Heilbronn conjecture == | |
| The '''Erdős–Heilbronn conjecture''' posed by [[Paul Erdős]] and [[Hans Heilbronn]] in 1964 states that <math>|2^\wedge A|\ge\min\{p,2|A|-3\}</math> if ''p'' is a prime and ''A'' is a nonempty subset of the field '''Z'''/''p'''''Z'''.<ref>Nathanson (1996) p.77</ref> This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994<ref>{{cite journal
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| | author = Dias da Silva, J. A.; Hamidoune, Y. O.
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| | title = Cyclic spaces for Grassman derivatives and additive theory
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| | journal = [[London Mathematical Society|Bulletin of the London Mathematical Society]]
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| | volume = 26
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| | year = 1994
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| | pages = 140–146
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| | doi = 10.1112/blms/26.2.140
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| | issue = 2}}</ref>
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| who showed that
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| :<math>|n^\wedge A|\ge\min\{p(F),\ n|A|-n^2+1\},</math>
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| 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 [[Imre Z. Ruzsa|I. Ruzsa]] in 1996,<ref name="Alon1996"/> Q. H. Hou and [[Zhi-Wei Sun]] in 2002,<ref>{{cite journal
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| | author = Hou, Qing-Hu; [[Zhi-Wei Sun|Sun, Zhi-Wei]]
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| | title = Restricted sums in a field
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| | journal = [[Acta Arithmetica]]
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| | volume = 102
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| | year = 2002
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| | issue = 3
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| | pages = 239–249
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| | mr = 1884717
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| | doi = 10.4064/aa102-3-3}}</ref>
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| and G. Karolyi in 2004.<ref>{{cite journal
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| | author = Károlyi, Gyula
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| | title = The Erdős–Heilbronn problem in abelian groups
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| | journal = [[Israel Journal of Mathematics]]
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| | volume = 139
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| | year = 2004
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| | pages = 349–359
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| | mr = 2041798
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| | doi = 10.1007/BF02787556}}</ref>
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| == Combinatorial Nullstellensatz ==
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| A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz.<ref name="Alon1999">{{cite journal
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| | author = [[Noga Alon|Alon, Noga]]
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| | url = http://www.math.tau.ac.il/~nogaa/PDFS/null2.pdf
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| | title = Combinatorial Nullstellensatz
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| | journal = [[Combinatorics, Probability and Computing]]
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| | volume = 8
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| | issue = 1–2
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| | year = 1999
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| | pages = 7–29
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| | mr = 1684621
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| | doi = 10.1017/S0963548398003411}}</ref> Let <math>f(x_1,\ldots,x_n)</math> be a polynomial over a field ''F''. Suppose that the coefficient of the monomial <math>x_1^{k_1}\cdots x_n^{k_n}</math> in <math>f(x_1,\ldots,x_n)</math> is nonzero and <math>k_1+\cdots+k_n</math> is the [[total degree]] of <math>f(x_1,\ldots,x_n)</math>. If <math>A_1,\ldots,A_n</math> are finite subsets of ''F'' with <math>|A_i|>k_i</math> for <math>i=1,\ldots,n</math>, then there are <math>a_1\in A_1,\ldots,a_n\in A_n</math> such that <math>f(a_1,\ldots,a_n)\not = 0 </math>.
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| 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,<ref>{{cite journal
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| | author = [[Noga Alon|Alon, Noga]]; Tarsi, Michael
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| | title = A nowhere-zero point in linear mappings
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| | journal = [[Combinatorica]]
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| | volume = 9
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| | year = 1989
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| | pages = 393–395
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| | mr = 1054015
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| | doi = 10.1007/BF02125351
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| | issue = 4}}</ref>
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| and developed by Alon, Nathanson and Ruzsa in 1995-1996,<ref name="Alon1996">{{cite journal
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| | author = [[Noga Alon|Alon, Noga]]; Nathanson, Melvyn B.; Ruzsa, Imre
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| | url = http://www.math.tau.ac.il/~nogaa/PDFS/anrf3.pdf
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| | title = The polynomial method and restricted sums of congruence classes
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| | journal = [[Journal of Number Theory]]
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| | volume = 56
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| | issue = 2
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| | year = 1996
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| | pages = 404–417
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| | mr = 1373563
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| | doi = 10.1006/jnth.1996.0029}}</ref>
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| and reformulated by Alon in 1999.<ref name="Alon1999"/>
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| ==References==
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| {{reflist|2}}
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| * {{cite book | editor1-last=Geroldinger | editor1-first=Alfred | editor2-last=Ruzsa | editor2-first=Imre Z. | others=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) | title=Combinatorial number theory and additive group theory | series=Advanced Courses in Mathematics CRM Barcelona | location=Basel | publisher=Birkhäuser | year=2009 | isbn=978-3-7643-8961-1 | zbl=1177.11005 }}
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| * {{cite book | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | year=1996 | isbn=0-387-94655-1 | zbl=0859.11003 }}
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| == External links ==
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| *{{cite journal | author = [[Zhi-Wei Sun|Sun, Zhi-Wei]] | title = An additive theorem and restricted sumsets | year = 2006 | pages = 1263–1276 | volume = 15 | issue = 6 | journal = Math. Res. Lett. , no. | arxiv = math.CO/0610981 }}
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| * [[Zhi-Wei Sun]]: [http://math.nju.edu.cn/~zwsun/EHLS.pdf On some conjectures of Erdős-Heilbronn, Lev and Snevily] ([[PDF]]), a survey talk.
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| *{{mathworld | urlname = Erdos-HeilbronnConjecture | title = Erdos-Heilbronn Conjecture}}
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| [[Category:Sumsets]]
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| [[Category:Additive combinatorics]]
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| [[Category:Additive number theory]]
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