CHAPTER 3

More properties of admissible arrays

In this chapter we prove a useful generalization of condition C (see Definition

2.3) and discuss a number of consequences.

THEOREM

3.1. Let L = {ieZ\li ra-hl} with the usual ordering be given

and let E denote a set with n elements. Assume that {#2 : Z — E | i G L} is an

admissible array on n symbols. If Bi := {0{(j) | j G Z}, assume that Bi C\Bi+i ^ 0

for 1 i m and write pi = \Bi\ (so Oi is periodic with minimal period pi).

Assume that Sj C L, 1 j fi, are pairwise disjoint subsets of L and that

L = Uj=i Sj- F°r each 3i 1 — 3 — M; assume that there is an integer rj 1 such

that for all i G Sj

gcd(pi-upi)\rj and gcd(pi,pi+i)\rj (3.1)

(If i = 1, the equation gcd(pi-i,pi)\rj is vacuous; and if i = m + 1, the equation

gcd(pi,Pi+i)\rj is vacuous.) Define Sj = \Sj\ and r = lcm({rj | 1 j

/J,}).

Then

we have

J2 Sj

= m + 1

and

j=l

TJ

PROOF. It is clear from the assumptions that Eq. (3.2) holds, that L — U?=i Sj

and that the sets Sj are pairwise disjoint. The point is to prove Eq. (3.3).

Because we assume that Bi f! B{+\ ^ 0 for 1 i m, there exist s^U G Z

such that

6i(si) = ei+1(U), lim. (3.4)

We define numbers 8i, 1 i m, and 77A, 0 A m, by

6i := Si - U, (3.5)

and

A

77A

:=^6i for 1 A m, r]0 := 0. (3.6)

2 = 1

For l A m + l , we define j \ , with 1 j \ JJL, by the equation

XeSjx f o r l A r a + l. (3.7)

10

(3.2)

(3.3)