关于加法组合的一些新问题  

SOME NEW PROBLEMS IN ADDITIVE COMBINATORICS

作　　者：孙智伟[1] Sun Zhi-Wei(Department of Mathematics,Nanjing University,Nanjing 210093)

机构地区：[1]南京大学数学系,南京210093

出　　处：《南京大学学报（数学半年刊）》2019年第2期134-155,共22页Journal of Nanjing University(Mathematical Biquarterly)

基　　金：Supported by the National Natural Science Foundation(grant 11571162)of China.

摘　　要：本文中研究加法组合中一些新问题。我们的问题主要涉及n个不同数(或加法Abel群的元)a1,...,an的使得诸ai+ai+1(或ai-ai+1)两两不同的置换或圆排列。对不等于25的奇素数幂次q=2n+1>13,我们证明有S={a^2:a∈Fq\{0}}中元的圆排列(a1,...,an)使得{a1+a2,...,an-1+an,an+a1}=S,这儿Fq指q元域。对无挠加法Abel群的n>3元有限子集A,我们证明有A的元素列举a1,...,an使得a1+2a2,a2+2a3,...,an-1+2an,an+2a1两两不同。我们还提出了30个未解决的猜想供进一步研究。In this paper we investigate some new problems in additive combinatorics.Our problems mainly involve permutations(or circular permutations) n distinct numbers(or elements of an additive abelian group) a1,…,an with adjacent sums ai+ai+1(or differences ai-ai+1) pairwise distinct.For an odd prime power q=2n+1> 13 with q≠25,we show that there is a circular permutation(a1,…,an) of the elements of S={a2:a∈Fq \ {0}} such that {a1+a2,…,an-1+an,an+a1}=S,where Fq denotes the field of order q.For any finite subset A of an additive torsion-free abelian group G with |A|=n> 3,we prove that there is a numbering a1,…,an of the elements of A such that a1+2 a2,a^2+2 a3,…,an-1+2an,an+2a1 are pairwise distinct.We also pose 30 open conjectures for further research.

关 键 词：加法组合 置换 素数 整除性 二次剩余 ABEL群 

分 类 号：O156[理学—数学]