THE RAMANUJAN JOURNAL, Vol. 3, No. 1 (1999), 73-81.

Continued fractions and unique additive partitions

DAVID J. GRABINER
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109
grabiner@math.lsa.umich.edu

Received April 29, 1997; Accepted April 23, 1998

Abstract. A partition of the positive integers into sets $A$ and $B$ {\it 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 {\it uniquely avoidable}. For any irrational $\alpha>1$, Chow and Long constructed a partition which avoids the numerators of all convergents of the continued fraction for $\alpha$, and conjectured that the set $S_{\alpha}$ 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 $\alpha$ has infinitely many partial quotients equal to $1$. We also construct the set $S_{\alpha}$ and show it is always uniquely avoidable.

Keywords: additive partition, best approximation, continued fraction, uniquely avoidable set

1991 Mathematics Classification: Primary 11P81; Secondary 05C40, 11J70