The Plus-Minus Davenport Constant of Finite Abelian Groups

Abstract

Let G be a finite abelian group, written additively. The Davenport constant, D(G), is the smallest positive number s such that any subset of the group G, with cardinality at least s, contains a non-trivial zero-subsum. We focus on a variation of the Davenport constant where we allow addition and subtraction in the non-trivial zero-subsum. This constant is called the plus-minus Davenport constant, D±(G). In the early 2000’s, Marchan, Ordaz, and Schmid proved that if the cardinality of G is less than or equal to 100, then the D±(G) is the floor of log2 n + 1, the basic upper bound, with few exceptions. The value of D±(G) is primarily known when the rank of G at most two and the cardinality of G is less than or equal to 100. In most cases, when D±(G) is known, D±(G)= floor(log2 |G|) + 1, with the exceptions of when G is a 3-group or a 5-group. We have studied a class of groups where the cardinality of G is a product of two prime powers. We look more closely to when the primes are 2 and 3, since the plus-minus Davenport constant of a 2-group attains the basic upper bound and while the plus-minus Davenport constant of a 3-group does not. To help us compute D±(G), we define the even plus-minus Davenport constant, De±(G), that guarantees a pm zero-subsum of even length. Let Cn be a cyclic group of order n. Then D(Cn) = n and D±(Cn) =floor( log2 n)+1. We have shown that De±(Cn) depends on whether n is even or odd. When n is even and not a power of 2, then De±(Cn) = floor(log2 n) + 2. When n = 2k , then De±(Cn) = floor(log2 n) + 1. The case when n is odd, De±(Cn) varies depending on how close n is to a power of 2. We have also shown that a subset containing the Jacobsthal numbers provides a subset of Cn that does not contain an even pm zero-subsum for certain values of n. When G is a finite abelian group, we provide bounds for De±(G). If D±(G) is known, then we given an improvement to the lower bound of De±(G). Additional improvements are shown when G is a direct sum an elementary abelian p-groups where p is prime. Then we compute the values of De±(Cr3 ) when 2 ≤ r ≤ 9 and provide an optimal lower bound for larger r. For the group C2 ⊕ Cr3 , D±(C2 ⊕ Cr3 ) = De±(Cr3 ). When r < 10, D±(C2 ⊕ Cr3 ) does not attain the basic upper bound. We conjecture that as r increases, D±(C2 ⊕ Cr3 ) will not attain the basic upper bound. Now, let G = Cq2 ⊕ Cr3 . We compute the values of D±(G) for general q and small r. In this case, we show that if D±(G) attains the basic upper bound then so does De±(G). We then look at the case when the cardinality of G is a product of two prime powers and show improvements on the lower bound by using the fractional part of log2 p of each prime. Furthermore, we compute the values of D±(G) when 100 < |G| ≤ 200, with some exceptions.