# IJPAM: Volume 10, No. 1 (2004)

**THE STRUCTURE OF COMPLEMENTARY SETS**

OF INTEGERS: A 3-SHIFT THEOREM

OF INTEGERS: A 3-SHIFT THEOREM

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