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)
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