On a conjecture of Lemke and Kleitman

• Autorzy: Xiangneng Zeng , Yuanlin Li , Pingzhi Yuan
• Języki publikacji: EN

Abstrakty

EN

Let G be a finite cyclic group of order n ≥ 2. Every sequence S over G can be written in the form $S = (n_1 g)· ... · (n_lg)$ where g ∈ G and $n_1, ..., n_l ∈ [1,ord(g)]$, and the index ind(S) of S is defined as the minimum of $(n_1 + ⋯ + n_l )/ord(g)$ over all g ∈ G with ord(g) = n. In this paper it is shown that any sequence S over G of length |S| ≥ n ≥ 5, 2 ∤ n, having an element with multiplicity at least n/3 has a subsequence T with ind(T) = 1. On the other hand, if n,d ≥ 2 are positive integers with d|n and $n > d²(d³-d²+d+1), we provide an example of a sequence S of length |S| ≥ n having an element with multiplicity l = n/d - d(d-1) - 1 such that S has no subsequence T with ind(T) = 1, giving a general counterexample to a conjecture of Lemke and Kleitman.

Kategorie tematyczne

• 11P70: Inverse problems of additive number theory, including sumsets
• 11B50: Sequences (mod m )

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 168
• Numer: 3
• Strony: 289-299

Daty

wydano

2015

Twórcy

autor

Xiangneng Zeng

• Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Guangzhou 510275, P.R. China

autor

Yuanlin Li

• Department of Mathematics and Statistics, Brock University, St. Catharines, ON, Canada L2S 3A1

autor

Pingzhi Yuan

• School of Mathematics, South China Normal University, Guangzhou 510631, P.R. China

Identyfikatory

DOI

10.4064/aa168-3-5