IJPAM: Volume 10, No. 1 (2004)

THE STRUCTURE OF COMPLEMENTARY SETS
OF INTEGERS: A 3-SHIFT THEOREM

Aviezri S. Fraenkel, Dalia Krieger
Department of Computer Science and Applied Mathematics
Weizmann Institute of Science
Rehovot 76100, ISRAEL
e-mail: fraenkel@wisdom.weizmann.ac.il
e-mail: daliak@wisdom.weizmann.ac.il

Abstract.Let be two irrational numbers satisfying . Then the sequences , , , are complementary over , thus satisfies: , (, the smallest positive integer not in the set ). Suppose that is an integer. Then for all .

We define the following generalization of sequences , : Let , and let be an arbitrary finite set. Let , , . Let . We show that no matter how we pick and , from some point on the shift sequence assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractal-like pattern, which we describe.

This work was motivated by a generalization of Wythoff game to piles.

Received: September 11, 2003

AMS Subject Classification: 05A17, 91A05

Key Words and Phrases: complementary sequences, integer part function, 3-shift theorem, Wythoff games

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 10
Issue: 1