On weighted zero-sum sequences

Abstract

Let G be a finite additive abelian group with exponent exp(G) = n > 1 and let A be a nonempty subset of {1, . . . , n − 1}. In this paper, we investigate the smallest positive integer m, denoted by sA(G), such that any sequence {ci} m i=1 with terms from G has a length n = exp(G) subsequence {ci j } n j=1 for which there are a1, . . . , an ∈ A such that n j=1 aici j = 0. In the case A = {±1}, we determine the asymptotic behavior of s {±1} (G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank,Combined with a lower bound of exp(G) + r i=1 ⌊log 2 ni⌋, where G ∼ = Zn 1 ⊕ · · · ⊕ Zn r with 1 < n1| · · · |nr, this determines s {±1} (G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems.Some additional more specific values and results related to s {±1} (G) are also computed.

Inverse results for weighted Harborth constants

“…In their study of the plus-minus weighted Erdős-Ginzburg-Ziv and Davenport constants Adhikari, Grynkiewicz, and Sun [3] established the following useful result. (1) If |S| > log 2 |G|, then S has a non-empty plus-minus weighted zero-subsum.…”

Number of Weighted Subsequence Sums with Weights in {1, –1}

“…It is not difficult to observe that for a finite abelian group G of the form G Š Z n 1˚Z n 2˚ ˚Z n r , 1 < n 1 j j n r , satisfying jGj > 2 .2 r 0 1 1C r 0 2 / , where r 0 D j¹i 2 ¹1; 2; : : : ; rº W 2 j n i ºj, and A D ¹1; 1º, our theorem along with some counter examples like those given in [2] (see also [4]) yields jGj C r X i D1 blog 2 n i c Ä E A .G/ Ä jGj C blog 2 jGjc: (8) This gives the exact value of E A .G/ when G is cyclic (thus giving another proof of the main result in [2]) and unconditional bounds in many cases.…”

“…However, we mention that when A D ¹1; 1º, finding the corresponding bounds for D A .G/ for a finite abelian group G and the exact value of D A .G/ when G is cyclic is not so difficult (see [2], [4]). Therefore, from the relation E A .G/ D D A .G/ C n 1;…”

Warning’s Second Theorem with restricted variables

“…A recent paper of DAGS [12] treats the analogous problem in any finite commutative p-group, with zero-sum subsequences replaced by generalized zero-sum subsequences in the sense of Section 4.2. As before, using Theorem 1.6 we get a quantitative refinement which also includes the inhomogeneous case.…”

“…It is interesting to compare this approach with the proof of Corollary 4.13b) given in [12]. Their argument proves the needed case of Theorem 1.5 by exploiting properties of binomial coefficients t d viewed as integer-valued polynomials and reduced modulo powers of p. In 2006 IPM lecture notes [30], R. Wilson proves Theorem 1.3 in this manner.…”