Stanley sequences with odd character

Given a set of integers containing no $3$-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. These sequences appear to have two distinct growth rates which dictate whether the sequences are structured or chaotic. Independent Stanley sequences are a “well-structured” class of Stanley sequences with two main parameters: the character $\lambda (A)$ and the repeat factor $\rho (A)$. Rolnick conjectured that for every $\lambda \in \mathbb{N}_0 \setminus \{1, 3, 5, 9, 11, 15 \}$, there exists an independent Stanley sequence $S(A)$ such that $\lambda (A) = \lambda$. This paper demonstrates that $\lambda (A) \notin \{ 1, 3, 5, 9, 11, 15 \}$ for any independent Stanley sequence $S(A)$.

Keywords

Stanley sequence, $3$-free set, arithmetic progression

2010 Mathematics Subject Classification

11B25

Full Text (PDF format)

Received 1 September 2017

Published 7 December 2018