Thresholds for zero-sums with small cross numbers in abelian groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913366044311552 |
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| author | Bushaw, Neal Hurlbert, Glenn |
| author_facet | Bushaw, Neal Hurlbert, Glenn |
| contents | For an additive group $Γ$ the sequence $S = (g_1, \ldots, g_t)$ of elements of $Γ$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_Γ$. 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 $Γ$. Call $S$ good if it contains a zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved that if $Γ$ is abelian then every length $|Γ|$ sequence of its elements is good, generalizing a 1989 result of Lemke and Kleitman that had proved an earlier conjecture of Erdős and Lemke. In 1989 Chung re-proved the Lemke and Kleitman result by applying a theorem of graph pebbling, and in 2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize Geroldinger's result. Here we use probabilistic theorems from graph pebbling to derive a threshold version of Geroldinger's theorem for abelian groups of a certain form. Specifically, we prove that if $p_1, \ldots, p_d$ are (not necessarily distinct) primes and $Γ_k$ has the form $\prod_{i=1}^d {\mathbb Z}_{p_i^k}$ then there is a function $τ=τ(k)$ (which we specify in Theorem 4) with the following property: if $t-τ\rightarrow\infty$ as $k\rightarrow\infty$ then the probability that $S$ is good in $Γ_k$ tends to 1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_03455 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Thresholds for zero-sums with small cross numbers in abelian groups Bushaw, Neal Hurlbert, Glenn Combinatorics 11P70 (Primary) 20K01, 60C05 (Secondary) For an additive group $Γ$ the sequence $S = (g_1, \ldots, g_t)$ of elements of $Γ$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_Γ$. 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 $Γ$. Call $S$ good if it contains a zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved that if $Γ$ is abelian then every length $|Γ|$ sequence of its elements is good, generalizing a 1989 result of Lemke and Kleitman that had proved an earlier conjecture of Erdős and Lemke. In 1989 Chung re-proved the Lemke and Kleitman result by applying a theorem of graph pebbling, and in 2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize Geroldinger's result. Here we use probabilistic theorems from graph pebbling to derive a threshold version of Geroldinger's theorem for abelian groups of a certain form. Specifically, we prove that if $p_1, \ldots, p_d$ are (not necessarily distinct) primes and $Γ_k$ has the form $\prod_{i=1}^d {\mathbb Z}_{p_i^k}$ then there is a function $τ=τ(k)$ (which we specify in Theorem 4) with the following property: if $t-τ\rightarrow\infty$ as $k\rightarrow\infty$ then the probability that $S$ is good in $Γ_k$ tends to 1. |
| title | Thresholds for zero-sums with small cross numbers in abelian groups |
| topic | Combinatorics 11P70 (Primary) 20K01, 60C05 (Secondary) |
| url | https://arxiv.org/abs/2309.03455 |