Continued Fractions and Unique Additive Partitions

A partition of the positive integers into sets A and B avoids a set S 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 > 1, Chow and Long constructed a partition which avoids the numerators of all convergents of the continued fraction for , and conjectured that the set S which this partition avoids is uniquely avoidable. We prove that the set of numerators of convergents is uniquely avoidable if and only if the continued fraction for has infinitely many partial quotients equal to 1. We also construct the set S and show that it is always uniquely avoidable.