|
|
| Line 1: |
Line 1: |
| In [[additive number theory]] and [[combinatorics]], a '''restricted sumset''' has the form
| | Myrtle Benny is how I'm called and I really feel comfy when individuals use the full title. Years in the past he moved to North Dakota and his family members enjoys it. Playing baseball is the pastime he will never quit doing. For many years I've been working as a payroll clerk.<br><br>Here is my blog [http://cs.sch.ac.kr/xe/index.php?document_srl=1203900&mid=Game13 cs.sch.ac.kr] |
| | |
| :<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>
| |
| | |
| 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''.
| |
| | |
| 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
| |
| | |
| :<math>P(x_1,\ldots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i),</math>
| |
| | |
| ''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>.
| |
| | |
| == 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<ref>Nathanson (1996) p.44</ref><ref name=GR1412>Geroldinger & Ruzsa (2009) pp.141–142</ref>
| |
| | |
| :<math>|A+B|\ge\min\{p,\ |A|+|B|-1\}.\,</math>
| |
| | |
| 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>
| |
| | |
| 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>
| |
| | |
| [[Kneser's theorem (combinatorics)|Kneser's theorem]] generalises this to finite abelian groups.<ref name=GR143>Geroldinger & Ruzsa (2009) p.143</ref>
| |
| | |
| == 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
| |
| | author = Dias da Silva, J. A.; Hamidoune, Y. O.
| |
| | title = Cyclic spaces for Grassman derivatives and additive theory
| |
| | journal = [[London Mathematical Society|Bulletin of the London Mathematical Society]]
| |
| | volume = 26
| |
| | year = 1994
| |
| | pages = 140–146
| |
| | doi = 10.1112/blms/26.2.140
| |
| | issue = 2}}</ref>
| |
| who showed that
| |
| | |
| :<math>|n^\wedge A|\ge\min\{p(F),\ n|A|-n^2+1\},</math>
| |
| | |
| 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
| |
| | author = Hou, Qing-Hu; [[Zhi-Wei Sun|Sun, Zhi-Wei]]
| |
| | title = Restricted sums in a field
| |
| | journal = [[Acta Arithmetica]]
| |
| | volume = 102
| |
| | year = 2002
| |
| | issue = 3
| |
| | pages = 239–249
| |
| | mr = 1884717
| |
| | doi = 10.4064/aa102-3-3}}</ref>
| |
| and G. Karolyi in 2004.<ref>{{cite journal
| |
| | author = Károlyi, Gyula
| |
| | title = The Erdős–Heilbronn problem in abelian groups
| |
| | journal = [[Israel Journal of Mathematics]]
| |
| | volume = 139
| |
| | year = 2004
| |
| | pages = 349–359
| |
| | mr = 2041798
| |
| | doi = 10.1007/BF02787556}}</ref>
| |
| | |
| == 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.<ref name="Alon1999">{{cite journal
| |
| | author = [[Noga Alon|Alon, Noga]]
| |
| | url = http://www.math.tau.ac.il/~nogaa/PDFS/null2.pdf
| |
| | title = Combinatorial Nullstellensatz
| |
| | journal = [[Combinatorics, Probability and Computing]]
| |
| | volume = 8
| |
| | issue = 1–2
| |
| | year = 1999
| |
| | pages = 7–29
| |
| | mr = 1684621
| |
| | 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>.
| |
| | |
| 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
| |
| | author = [[Noga Alon|Alon, Noga]]; Tarsi, Michael
| |
| | title = A nowhere-zero point in linear mappings
| |
| | journal = [[Combinatorica]]
| |
| | volume = 9
| |
| | year = 1989
| |
| | pages = 393–395
| |
| | mr = 1054015
| |
| | doi = 10.1007/BF02125351
| |
| | issue = 4}}</ref>
| |
| and developed by Alon, Nathanson and Ruzsa in 1995-1996,<ref name="Alon1996">{{cite journal
| |
| | author = [[Noga Alon|Alon, Noga]]; Nathanson, Melvyn B.; Ruzsa, Imre
| |
| | url = http://www.math.tau.ac.il/~nogaa/PDFS/anrf3.pdf
| |
| | title = The polynomial method and restricted sums of congruence classes
| |
| | journal = [[Journal of Number Theory]]
| |
| | volume = 56
| |
| | issue = 2
| |
| | year = 1996
| |
| | pages = 404–417
| |
| | mr = 1373563
| |
| | doi = 10.1006/jnth.1996.0029}}</ref>
| |
| and reformulated by Alon in 1999.<ref name="Alon1999"/>
| |
| | |
| ==References==
| |
| {{reflist|2}}
| |
| * {{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 }}
| |
| * {{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 }}
| |
| | |
| == External links ==
| |
| *{{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 }}
| |
| * [[Zhi-Wei Sun]]: [http://math.nju.edu.cn/~zwsun/EHLS.pdf On some conjectures of Erdős-Heilbronn, Lev and Snevily] ([[PDF]]), a survey talk.
| |
| *{{mathworld | urlname = Erdos-HeilbronnConjecture | title = Erdos-Heilbronn Conjecture}}
| |
| | |
| [[Category:Sumsets]]
| |
| [[Category:Additive combinatorics]]
| |
| [[Category:Additive number theory]]
| |
Myrtle Benny is how I'm called and I really feel comfy when individuals use the full title. Years in the past he moved to North Dakota and his family members enjoys it. Playing baseball is the pastime he will never quit doing. For many years I've been working as a payroll clerk.
Here is my blog cs.sch.ac.kr