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## Acta Arithmetica

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## On a conjecture of Lemke and Kleitman

### Volume 168 / 2015

Acta Arithmetica 168 (2015), 289-299 MSC: Primary 11B50; Secondary 11P70. DOI: 10.4064/aa168-3-5

#### Abstract

Let $G$ be a finite cyclic group of order $n\ge 2$. Every sequence $S$ over $G$ can be written in the form $S = (n_1g)\cdot \ldots \cdot (n_lg)$ where $g \in G$ and $n_1, \ldots , n_l \in [1, \mathop{\rm ord} (g)]$, and the index $\mathop{\rm ind} (S)$ of $S$ is defined as the minimum of $(n_1 +\cdots + n_l )/ \mathop{\rm ord} (g)$ over all $g\in G$ with $\mathop{\rm ord} (g) = n$. In this paper it is shown that any sequence $S$ over $G$ of length $|S| \ge n \ge 5$, $2 \nmid n$, having an element with multiplicity at least ${n/3}$ has a subsequence $T$ with $\mathop{\rm ind} (T ) = 1$. On the other hand, if $n, d\ge 2$ are positive integers with $d\,|\,n$ and $n>d^2(d^3-d^2+d+1)$, we provide an example of a sequence $S$ of length $|S| \ge n$ having an element with multiplicity $l={n/d}-d(d-1)-1$ such that $S$ has no subsequence $T$ with $\mathop{\rm ind} (T ) = 1$, giving a general counterexample to a conjecture of Lemke and Kleitman.

#### Authors

• Xiangneng ZengSino–French Institute of
Nuclear Engineering and Technology
Sun Yat-Sen University
Guangzhou 510275, P.R. China
e-mail
• Yuanlin LiDepartment of Mathematics and Statistics
Brock University
St. Catharines, ON, Canada L2S 3A1
e-mail
• Pingzhi YuanSchool of Mathematics
South China Normal University
Guangzhou 510631, P.R. China
e-mail

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