On the index-\(r\)-free sequences over finite cyclic groups. (English) Zbl 1428.11027
Summary: Let \(C_n\) be a finite cyclic group of order \(n \geq 2\). Every sequence \(S\) over \(C_n\) can be written in the form \(S = (n_1 g), \dots,(n_l g)\) where \(g \in C_n\) and \(n_1, \dots, n_l \in [1, \mathrm{ord}(g)]\), and the index \(\mathrm{ind}(S)\) of \(S\) is defined as the minimum of \((n_1 + \cdots + n_l) / \mathrm{ord}(g)\) over all \(g \in C_n\) with \(\mathrm{ord}(g) = n\). Let \(d > 1\) and \(r \geq 1\) be any fixed integers. We prove that, for every sufficiently large integer \(n\) divisible by \(d\), there exists a sequence \(S\) over \(C_n\) of length \(| S | \geq n + n / d + O(\sqrt{n})\) having no subsequence \(T\) of index \(\mathrm{ind}(T) \in [1, r]\), which has substantially improved the previous results in this direction.
MSC:
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 11B50 | Sequences (mod \(m\)) |
| 20K01 | Finite abelian groups |
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