Continued Fractions and Unique Additive Partitions
David J. Grabiner
A partition of the positive integers into sets $A$ and $B$ avoids a set $S\subset{\mathbb N}$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is uniquely avoidable. For any irrational $\alpha \gt 1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which this partition avoided was uniquely avoidable. We prove that the set of numerators of convergents is uniquely avoidable if and only if the continued fraction for $\alpha$ has infinitely many partial quotients equal to 1. We also construct the set $S_\alpha$ and show that it is always uniquely avoidable.