THE RAMANUJAN JOURNAL, Vol. 3, No. 1 (1999), 55-72.

Additive partitions and continued fractions

TIMOTHY Y. CHOW
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109
tchow@umich.edu

CHRISTOPHER D. LONG
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
clong@math.rutgers.edu

Received April 24, 1997; Accepted September 23, 1997

Abstract. A set $S$ of positive integers is {\it avoidable} if there exists a partition of the positive integers into two disjoint sets such that no two distinct integers 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 broaden the class of avoidable sets by establishing a previously unnoticed connection with the elementary theory of continued fractions.

Keywords: combinatorial number theory, additive number theory, Beatty's theorem, intermediate fractions, avoidable sets

1991 Mathematics Classification: Primary 11J70; Secondary 05A17, 11B75, 11P81