  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.
