[1]陈永高.Erdös-Turán猜想与相关的问题[J].南京师大学报(自然科学版),2013,36(04):1.
 Chen Yonggao.The Erdös-Turán Conjecture and Related Topics[J].Journal of Nanjing Normal University(Natural Science Edition),2013,36(04):1.
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Erdös-Turán猜想与相关的问题
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《南京师大学报(自然科学版)》[ISSN:1001-4616/CN:32-1239/N]

卷:
第36卷
期数:
2013年04期
页码:
1
栏目:
出版日期:
2013-12-31

文章信息/Info

Title:
The Erdös-Turán Conjecture and Related Topics
作者:
陈永高
南京师范大学数学科学学院与数学研究所,江苏 南京 210023
Author(s):
Chen Yonggao
School of Mathematical Sciences and Institute of Mathematics,Nanjing Normal University,Nanjing 210023,China
关键词:
Erdös-Turán猜想表示函数加法基
Keywords:
Erdös-Turán conjecturerepresentation functionadditive basis
分类号:
O156.1
文献标志码:
A
摘要:
设N是所有非负整数所成的集合,对于AN,用R(A,n)表示方程n=a+b,a,b∈A的解数.著名的Erdös-Turán猜想为:如果对所有非负整数n,总有R(A,n)≥1,则R(A,n)一定无界.本文简单介绍了Erdös-Turán猜想的进展,同时证明了Erdös-Turán猜想在有理数域上关于加法和乘法(乘法不含零)均不成立.如关于乘法,本文证明了如下结论:存在非零有理数集的一个子集A,使得每个非零有理数均可以唯一地(不考虑次序)表成A中两个数的乘积.最后,本文提出了7个未解决的问题供进一步研究.
Abstract:
Let N be the set of all nonnegative integers.For any subset A of N,let R(A,n)be the number of solutions to the equation n=a+b,a,b∈A.If R(A,n)≥1 for all integers n≥0,then A is called a basis of N.The well known Erdös-Turán conjecture says that if A is a basis of N,then R(A,n)cannot be bounded.In this paper,we give a brief review on the Erdös-Turán conjecture.We also prove that the Erdös-Turán conjecture is false in rational number field on both addition and multiplication.For example,the following result is proved:There exists a subset A of nonzero rational numbers such that every nonzero rational number α can be uniquely(neglecting the order)represented as a product of two elements of A.Finally,seven open questions are posed for further research.

参考文献/References:

[1] Erdös P,Turán P.On a problem of Sidon in additive number theory and on some related problems[J].J Lond Math Soc,1941,16:212-215.
[2]Erdös P.On a problem of Sidon in additive number theory[J].Acta Sci Math(Szeged),1954,15:255-259.
[3]Dubickas A.Additive bases of positive integers and related problems[J].Unif Distrib Theory,2008,3:81-90.
[4]Ruzsa I Z.A just basis[J].Monatsh Math,1990,109:145-151.
[5]Tang M.A note on a result of Ruzsa[J].Bull Aust Math Soc,2008,77:91-98.
[6]Tang M.A note on a result of Ruzsa,II[J].Bull Aust Math Soc,2010,82:340-347.
[7]Chen Y G,Yang Q H.Ruzsa’s theorem on Erdös and Turán conjecture[J].European J Combin,2013,34:410-413.
[8]Nathanson M B.Unique representation bases for integers[J].Acta Arith,2003,108:1-8.
[9]Grekos G,Haddad L,Helou C,et al.On the Erdös-Turán conjecture[J].J Number Theory,2003,102:339-352.
[10]Borwein P,Choi S,Chu F.An old conjecure of Erdös-Turán on additive bases[J].Math Comp,2005,75:475-484.
[11]Erdös P.On the multiplicative representation of integers[J].Israel J Math,1964,2:251-261.
[12]Tang M,Chen Y G.A basis of Zm[J].Colloq Math,2006,104:99-103.
[13]Tang M,Chen Y G.A basis of Zm,Ⅱ[J].Colloq Math,2007,108:141-145.
[14]Chen Y G.The analogue of Erdös-Turán conjecture in Zm[J].J Number Theory,2008,128:2 573-2 581.
[15]Chen Y G.On the Erdös-Turán conjecture[J].C R Acad Sci Paris,Ser I,2012,350:933-935.
[16]Yang Q H.A generalization of Chen’s theorem on the Erdös-Turán conjecture[J].Int J Number Theory,2013(9):1 683-1 686.
[17]Chen Y G,Sun T.The difference basis and bi-basis of Zm,Ⅱ[J].Colloq Math,2007,108:141-145.
[14]Chen Y G.The analogue of Erdös-Turán conjecture in Zm.J Number Theory,2010,130:716-726.
[18]Dubickas A,emetulskis G.On polynomials with flat squares[J].Acta Arith,2011,146:247-255.
[19]Dubickas A.A basis of finite and infinite sets with small representation[J].The Electronic J Combin,2012,19:R6.
[20]Haddad L,Helou C.Bases in some additive groups and the Erdös-Turán conjecture[J].J Combin Theory Ser A,2004,108:147-153.
[21]Nathanson M B.Generalized additive bases,König’s lemma,and the Erdös-Turán conjecture[J].J Number Theory,2004,106:70-78.
[22]Pǔs V.On multiplicative bases in abelian groups[J].Czech Math J,1991,41:282-287.
[23]Cilleruelo J,Ru J.On a question of Sárkozy and Ss for bilinear forms[J].Bull London Math Soc,2009,41:274-280.
[24]Moser L.An application of generating series[J].Math Mag,1962,35:37-38.

备注/Memo

备注/Memo:
收稿日期:2013-09-20.
基金项目:国家自然科学基金(11371195).
通讯联系人:陈永高,特聘教授,博士生导师,研究方向:数论.E-mail:ygchen@njnu.edu.cn
更新日期/Last Update: 2013-12-30