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 $\alpha>1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ 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 $\alpha$ has infinitely many partial quotients equal to 1. We also construct the set $S_\alpha$ and show that it is always uniquely avoidable.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 1997
 DOI:
 10.48550/arXiv.math/9704220
 arXiv:
 arXiv:math/9704220
 Bibcode:
 1997math......4220G
 Keywords:

 Mathematics  Combinatorics