Additive partitions and continued fractions (Q1288804)

If \(A_1\) is a subset of \(N\), let \(A_2\) denote the complement of \(A_1\). Let \(S\) be a subset of \(N\) such that the sum of any two elements of \(A_i\) is not in \(S\), where \(i= 1,2\) respectively. Then say that \(S\) is avoided by \(\{A_1, A_2\}\). The authors' main result is the following: if \(\alpha\) is an irrational number such that \(1< \alpha< 2\), \(A_1= A_\alpha\) is a certain subset of \(N\) that depends on \(\alpha\), and \(S_\alpha\) is the subset of \(N\) that is avoided by \(\{A_1, A_2\}\), then \(S_\alpha\) contains all numerators of continued fraction convergents of \(\alpha\).