The Weighted Davenport Constant of a Group and a Related Extremal Problem

Niranjan Balachandran,Eshita Mazumdar

doi:10.37236/7996

Abstract

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the following holds: Given an arbitrary sequence $(x_1,\ldots,x_k)$ with $x_i\in G$, there exists a non-empty subsequence $(x_{i_1},\ldots,x_{i_t})$ along with $a_{j}\in A$ such that $\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a natural new extremal problem that arises from the study of $D_A(G)$: For an integer $k\ge 2$, determine $f^{(D)}_G(k):=\min\{|A|: D_A(G)\le k\}$ (if the problem posed makes sense). It turns out that for $k$ 'not-too-small', this is a well-posed problem and one of the most interesting cases occurs for $G=\mathbb{Z}_p$, the cyclic group of prime order, for which we obtain near optimal bounds for all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to constants) bounds for $k=2,4$.

Highlights

Suppose a < b are positive integers
Proof. Before we start with the proof of the 3rd part of this theorem, we shall state a reformulation of what we seek: For any k 1, to find an upper bound for f (D)(p, 2k), it suffices to construct a set A ⊂ Z∗p of the requisite size such that for any α1, . . . , αk−1, β1, . . . , βk−1 ∈ Z∗p, Z∗p
One may frame the problem of obtaining an upper bound for f (D)(p, 2k − 1) by constructing a set A such that for any α1 . . . , αk, β1, . . . , βk−1 ∈ Z∗p such that

Summary

Introduction

Suppose a < b are positive integers. By [a, b] we denote the set {a, a + 1, . . . , b} ⊂ N. The Davenport Constant D(G), introduced by Rogers [12], is defined as the smallest k such that every G-sequence of length k contains a non-trivial zero-sum subsequence As it turns out, the Davenport constant is an important invariant of the ideal class group of the ring of integers of an algebraic number field (see [11] for more details). The focal point of this paper stems from a natural extremal problem in light of the known results on the Davenport constant of a group. For G = Zn, we shall write f (D)(n, k) := fG(D)(k) for convenience As it turns out, the nature of this extremal problem of determining fG(D)(k) is most interesting for the case when G is a cyclic group of prime order, and in that case, we establish the following bounds.

Preliminaries

Proofs of theorem 1 and theorem 2

Proof of theorem 2

A B shall denote the set

Concluding Remarks