Character Values of Stanley Sequences

TL;DR

This paper investigates the structure of Stanley sequences, especially character values of independent sequences, and proves that almost all even integers can be realized as characters, showing a dense set of such values.

Contribution

It significantly extends the known set of achievable character values for independent sequences, proving that all even integers except those congruent to 244 mod 486 can be characters.

Findings

All even integers not congruent to 244 mod 486 are achievable as characters.

The set of character values has a positive lower density.

The result improves upon previous limitations to base 3 representations.

Abstract

Stanley and Odlyzko proposed a method for greedily constructing sets with no 3-term arithmetic progressions. It is conjectured that there is a dichotomy between such sequences: those that have a periodic structure as the sequence satisfies certain recurrence relations while others appear to be chaotic. One large class of sequences that have these periodic behaviors are known as independent sequences that have two parameters, a character and a growth factor. It was conjectured by Rolnick that all but a finite set of integers can be achieved as characters of a independent sequences. Previously the only large class of integers known to be characters where those with base 3 representations consisting solely of the digits 0 and 2. This paper dramatically improves this result by demonstrating that all even integers not congruent to 244 mod 486 can be achieved as characters, therefore demonstrating that the set of all characters has a positive lower density.