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For an additive group $\Gamma$ the sequence $S = (g_1, \ldots, g_t)$ ofelements of $\Gamma$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_\Gamma$.The cross number of $S$ is defined to be the sum $\sum_{i=1}^k 1/|g_i|$, where$|g_i|$ denotes the order of $g_i$ in $\Gamma$. Call $S$ good if it contains azero-sum subsequence with cross number at most 1. In 1993, Geroldinger provedthat if $\Gamma$ is abelian then every length $|\Gamma|$ sequence of itselements is good, generalizing a 1989 result of Lemke and Kleitman that hadproved an earlier conjecture of Erd\H{o}s and Lemke. In 1989 Chung re-provedthe Lemke and Kleitman result by applying a theorem of graph pebbling, and in2005, Elledge and Hurlbert used graph pebbling to re-prove and generalizeGeroldinger's result. Here we use probabilistic theorems from graph pebbling toderive a sharp threshold version of Geroldinger's theorem for abelian groups ofa certain form. Specifically, we prove that if $p_1, \ldots, p_d$ are (notnecessarily distinct) primes and $\Gamma_k$ has the form $\prod_{i=1}^d{\mathbb Z}_{p_i^k}$ then there is a function $\tau=\tau(k)$ (which we specifyin Theorem 4) with the following property: if $t-\tau\rightarrow\infty$ as$k\rightarrow\infty$ then the probability that $S$ is good in $\Gamma_k$ tendsto 1, while if $\tau-t\rightarrow\infty$ then that probability tends to 0.

Thresholds for zero-sums with small cross numbers in abelian groups