On the Cross Numbers of Minimal Zero Sequences in Selected Cyclic Groups

In this paper, we deal with the set of possible cross numbers for two cyclic groups: Z2pn and Zpq. The first section completely characterizes this set for Z2pn . Next, we create a family of minimal zero sequences in Zpq. The following section utilizes these sequences to completely determine the structure of W (Zpq) for p sufficiently larger than q. This section additionally gives stability conditions on p and q for determining the structure of portions of W (Zpq), and also asymptotic analysis of the number of holes for fixed q and varying p. The last section determines the location of the two gaps of largest omitted cross numbers (whenever two exist) for any p, q odd primes, other than twin primes. A nonempty sequence S = {g1, . . . , gn} of not necessarily distinct elements of an additive group G is called a zero sequence if ∑n i=1 gi = 0. A zero sequence with no proper nonempty zero subsequence is called a minimal zero sequence. If a sequence contains no zero subsequence, it is known as zero-free. We define U(G) = {T | T is a minimal zero sequence in G}. Several constants can be extracted from such sequences, of which we mention a few. The Davenport constant, D(G) is the maximum length of a minimal zero sequence in G. The Davenport constant is at most the order of G, and in the case of cyclic groups it indeed attains that value. We additionally define the cross number of a sequence as