Characters of Independent Stanley Sequences

TL;DR

This paper proves that all nonnegative integers except 1, 3, 5, 9, 11, and 15 can be realized as the character of an independent Stanley sequence, resolving a conjecture and advancing understanding of their structure.

Contribution

It establishes that every nonnegative integer outside a specific small set can be the character of an independent Stanley sequence, confirming a conjecture by Rolnick.

Findings

Every nonnegative integer except 1, 3, 5, 9, 11, 15 is attainable as a character.

The result advances the classification of independent Stanley sequences.

The paper resolves a previously open conjecture about sequence characters.

Abstract

Odlyzko and Stanley introduced a greedy algorithm for constructing infinite sequences with no 3-term arithmetic progressions when beginning with a finite set with no 3-term arithmetic progressions. The sequences constructed from this procedure are known as Stanley sequences and appear to have two distinct growth rates which dictate whether the sequences are structured or chaotic. A large subclass of sequences of the former type is independent sequences, which have a self-similar structure. An attribute of interest for independent sequences is the character. In this paper, building on recent progress, we prove that every nonnegative integer $\lambda\not\in\{1,3,5,9,11,15\}$ is attainable as the character of an independent Stanley sequence, thus resolving a conjecture of Rolnick.