Additive partitions and continued fractions

A set S of positive integers is avoidable if there exists a partition of th e positive integers into two disjoint sets such that no two distinct intege rs from the same set sum to an element of S. Much previous work has focused on proving the avoidability of very special sets of integers. We vastly br oaden the class of avoidable sets by establishing a previously unnoticed co nnection with the elementary theory of continued fractions.