CONTINUED FRACTIONS AND UNIQUE ADDITIVE PARTITIONS

DAVID J. GRABINER

Abstract. 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 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.