Continued Fractions and Unique Additive Partitions

Abstract

A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational $α>1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $α$, and conjectured that the set $S_α$ 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 $α$ has infinitely many partial quotients equal to 1. We also construct the set $S_α$ and show that it is always uniquely avoidable.