Zero-Sum Problems in Finite Abelian Groups and Affine Caps

Abstract

For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form G = C r n , but they respect the structure of the group. In particular, we show s(C 4 n) ≥ 20n − 19 for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.

An application of coding theory to estimating Davenport constants

“…For example, η(C 6 3 ) was determined only recently [27], despite the fact that the problem of determining η(C r 3 ) is fairly popular (see [10] for a detailed outline of several problems, and their respective history, that are equivalent to determining η(C It is known, in particular by the work of Delorme, Ordaz, and Quiroz [8], that the invariants s ≤x (G) can be used to derive upper bounds for D j (G). More specifically, we have (this is Lemma 2.4 in [14])…”

Inverse zero-sum problems

“…Then, by definition, there exists some c(G) ∈ N such that s(G) = c(G)(n − 1) + 1. Moreover, a simple argument shows that Property C holds (see [24,Section 7]) and that η(G) = (c(G) − 1)(n − 1) + 1 (see [13,Lemma 2.3]). For r = 1 we have c(G) = 2 and for r = 2 we have c(G) = 4 (see Theorem 2.4).…”

“…For r = 1 we have c(G) = 2 and for r = 2 we have c(G) = 4 (see Theorem 2.4). In the case of higher ranks bounds for c(G) were given by N. Alon and M. Dubiner (see [1]) and then by Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin and L. Rackham (see [32,15,13,12]). We make use of the simple fact that η(C r 2 ) = 2 r and s(C r 2 ) = 2 r + 1 (see [30,Corollary 5.7.6]).…”

“…We make use of the simple fact that η(C r 2 ) = 2 r and s(C r 2 ) = 2 r + 1 (see [30,Corollary 5.7.6]). It follows from the very definition that C r 2 satisfies Property D, and a straightforward argument shows that C r 3 satisfies Property D (see [13,Lemma 2.3.3] and the subsequent discussion). However, in general only very little is known about Property D. If r = 2 and n ∈ {2, 3, 5, 7}, then G has Property D by [37], and for first results in groups of higher rank we refer to [25].…”

Roth’s theorem in ℤ4n

“…Indeed, for ‫ޚ‬ n 3 (or, more generally, any abelian group of odd order and bounded exponent), Roth's original argument simplifies considerably to give the following result, which is qualitatively due to Brown and Buhler [1984]. The question of what the true bounds on |A| are arises in many different studies [Frankl et al 1987;Yekhanin and Dumer 2004;Edel 2004;Edel et al 2007] and improving the bound is a well known open problem, as reported in [Green 2005;Croot and Lev 2007;Tao 2008, Section 3.1]; the closest anyone has come is in ]. While we are not able to make progress on this question, it is the purpose of this paper to show an improvement for a different class of groups.…”