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Two conjectures on an addition theorem

100%

Zeng X., Yuan P.

Acta Arithmetica

2011

tom 148

nr 4

395-411

On a conjecture of Lemke and Kleitman

81%

Zeng X., Li Y., Yuan P.

Acta Arithmetica

2015

tom 168

nr 3

289-299

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.

Ograniczanie wyników

2 Acta Arithmetica

2 Yuan P.

2 Zeng X.

1 Li Y.

1 2015

1 2011

Wyniki wyszukiwania

Two conjectures on an addition theorem

On a conjecture of Lemke and Kleitman