Some Zero Sum Problems in combinatorial number theory
Some Zero Sum Problems in combinatorial number theory
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发表时间：
2011
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通讯作者：
B. K. Moriya
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作者：
B. K. Moriya
本论文包括三个结果，每个结果都在单独的章节中讨论。正如标题所示，第一章是介绍性的。其他三章专门讨论三个不同的问题。下面简单介绍一下我们的结果。 1. 设 G 为任意秩为 r 的有限阿贝尔群，具有不变量 n1、n2、····、nr。换句话说，G = Zn1 ⊕Zn2 ⊕···⊕Znr 其中 ni 是满足 1 < n1|n2| 的整数· · · |编号。群 G 的达文波特常数定义为最小的正整数 t，使得 G 的每个长度为 t 的元素序列都具有非空零和子序列。 Śliwa 推测，D(G) ≤ Σi=1 ni。按照这个猜想，我们得到了 G 的达文波特常数 D(G) 的以下上限，D(G) ≤ nr+nr−1+(c(3)−1)nr−2+(c(4)−1)nr−3+···+(c(r)−1)n1+1，其中 c(i) 是各个 i 的 Alon-Dubiner 常数 [10]。我们还将给出达文波特常数在二次筛中的应用。 2. 令G 为有限交换群，其中exp(G) = e。设 s(G)（分别为 η(G)）为最小正整数 t，其属性为 G 的元素长度为 t 的任何序列 S 都包含 S 的 e 项子序列（分别为长度至多为 e 的非空子序列）且其和为零。对于等级最多为 2 的组，该常数已完全确定（参见[45]）。考虑等级大于 2 的组的问题，得出了这个结果。我们的问题是在 n、m 和 r 的一些约束下确定 s(C nm) 的值。令n、m和r为正整数且m ≥ 3。此外，η(C m) = ar(m− 1) + 1，对于某个常数ar取决于r并且n是大于或等于的固定整数，mr(c(r)m− ar(m− r) +m− 3)(m− 1)− (m+ 1) + (m+ 1)(ar + 1) m(m+ 1)(ar + 1) 1) 和 s(C n) = (ar +1)(n− 1) + 1。在上述 n 下界中，c(r) 是 Alon-Dubiner 常数。那么 s(C nm) = (ar + 1)(nm− 1) + 1。 3. 给定 n 阶阿贝尔群 G 和整数的有限非空子集 A，权重为 A 的 G 的达文波特常数（记为 DA(G)）被定义为最小正整数 t，使得每个具有 xi ∈ G 的序列 (x1, · · · , xt) 都有一个非空子序列(xj1 , · · · , xjl) 和 ai ∈ A 使得 Σl i=1 aixji = 0。类似地，EA(G) 被定义为最小正整数 t，使得 G 元素长度为 t 的每个序列 (x1, · · · , xt) 都有一个子序列 (xj1 , · · · , xjn) 使得 Σn i=1 aixji = 0，对于某些 ai ε A。当G的阶数为n时，认为A是{1,···,n−1}的非空子集。如果 G 是循环群 Z/nZ，我们分别用 EA(n) 和 DA(n) 表示 EA(G) 和 DA(G)。在这里，我们扩展了 Adhikari 等人的一篇文章中的一些结果。 [5] 并确定 DRn(n) 和 ERn(n) 的界限，其中 Rn = {x2 : x ∈ (Z/nZ)*} 且 (Z/nZ) 是一组以 n 为模的单位。我们遵循 [5] 中的一些论点，并使用 Yuan 和 Zeng [79] 的最新结果，这是由 I. Chowla [24] 和 Kneser 定理 [52] 得出的定理。
This thesis comprises of three results each of which dealt in separate chapters. First chapter is of introductory nature, as the title suggest. And the other three chapters are devoted to three different problems. Following is a brief introduction to our results. 1. Let G be any finite abelian group of rank r with invariants n1, n2, · · · , nr. In other words, G = Zn1 ⊕Zn2 ⊕· · ·⊕Znr where ni’s are integers satisfying 1 < n1|n2| · · · |nr. The Davenport constant of a group G is defined as the smallest positive integer t such that every sequence of length t of elements of G has a non-empty zero-sum subsequence. It has been conjectured by Śliwa that, D(G) ≤ ∑i=1 ni. Thinking in the direction of this conjecture we have obtained the following upper bound on Davenport constant D(G), of G, D(G) ≤ nr+nr−1+(c(3)−1)nr−2+(c(4)−1)nr−3+· · ·+(c(r)−1)n1+1, where c(i)’s are Alon-Dubiner constants [10] for respective i’s. Also we shall give an application of Davenport’s constant to Quadratic sieve. 2. LetG be a finite abelian group with exp(G) = e. Let s(G) (respectively, η(G)) be the minimal positive integer t with the property that any sequence S of length t of elements ofG contains an e-term subsequence (respectively, a non-empty subsequence of length at most e) of S with sum zero. For the group of rank at most two this constant has been determined completely (see [45]). Looking at the problem for groups of rank greater that 2 gave rise to this result. Our problem is to determine value of s(C nm) under some constraints on n,m, and r. Let n,m and r be positive integers andm ≥ 3. Furthermore, η(C m) = ar(m− 1) + 1, for some constant ar depending on r and n is a fixed integer greater than or equal to, mr(c(r)m− ar(m− r) +m− 3)(m− 1)− (m+ 1) + (m+ 1)(ar + 1) m(m+ 1)(ar + 1) and s(C n) = (ar +1)(n− 1) + 1. In the above lower bound on n, c(r) is the Alon-Dubiner constant. Then s(C nm) = (ar + 1)(nm− 1) + 1. 3. Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weight A, denoted by DA(G), is defined to be the least positive integer t such that every sequence (x1, · · · , xt) with xi ∈ G has a non-empty subsequence (xj1 , · · · , xjl) and ai ∈ A such that ∑l i=1 aixji = 0. Similarly, EA(G) is defined to be the least positive integer t such that every sequence (x1, · · · , xt) of length t of elements ofG has a subsequence (xj1 , · · · , xjn) such that ∑n i=1 aixji = 0, for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1, · · · , n− 1}. If G is the cyclic group Z/nZ we denote EA(G) and DA(G) by EA(n) and DA(n) respectively. Here we extend some results in an article of Adhikari et al. [5] and determine bounds for DRn(n) and ERn(n), where Rn = {x2 : x ∈ (Z/nZ)∗} and (Z/nZ) is a group of units modulo n. We follow some line of arguments in [5] and use a recent result of Yuan and Zeng [79], a theorem due to I. Chowla [24] and Kneser’s theorem [52].