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Some new results about a conjecture by Brian Alspach
Archiv der Mathematik ( IF 0.5 ) Pub Date : 2020-08-29 , DOI: 10.1007/s00013-020-01507-7
S. Costa , M. A. Pellegrini

In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is possible to find an ordering $(a_1,\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\sum_{j=1}^i a_j$, $i=1,\ldots,k$, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $k\leq 11$ in cyclic groups of prime order. Here, we extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\mathbb{Z}_n$. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset $A$ of $\mathbb{Z}_p\setminus\{0\}$, where $p$ is a prime, there exists an ordering of the elements of $A$ such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based on the Alon's combinatorial Nullstellensatz, we prove the validity of such conjecture for subsets $A$ of size $12$.

中文翻译:

Brian Alspach 猜想的一些新结果

在本文中,我们考虑以下由 Brian Alspach 提出的关于有限循环群中部分和的猜想:给定 $\mathbb{Z}_n\setminus \{0\}$ 的子集 $A$,其大小为 $k$ 这样$\sum_{z\in A} z\not= 0$,可以找到$A$元素的排序$(a_1,\ldots,a_k)$使得部分和$s_i=\ sum_{j=1}^i a_j$, $i=1,\ldots,k$, 非零且成对不同。已知这个猜想对于素数阶循环群中大小为 $k\leq 11$ 的子集是正确的。在这里,我们将这样的结果扩展到任何无扭阿贝尔群,因此,我们在 $\mathbb{Z}_n$ 中提供渐近结果。我们还考虑了一个相关的猜想,最初由 Ronald Graham 提出:给定 $\mathbb{Z}_p\setminus\{0\}$ 的子集 $A$,其中 $p$ 是素数,$A$ 的元素存在一个排序,使得部分和都是不同的。使用 Hicks、Ollis 和 Schmitt 开发的方法,基于 Alon 的组合 Nullstellensatz,我们证明了这种猜想对于大小为 $12$ 的子集 $A$ 的有效性。
更新日期:2020-08-29
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