On continued fractions and diophantine approximation in power ...

156 W. M. Schmidt
complete denominators αn, αn+t have
αn+t = [(B/D)C q
1
, (D/B)Cq
2
, . . .] = (B/D)[Cq 1 , Cq 2 , . . .] = (B/D)αq n.
Thus αn+t, α q n are connected by a fractional linear transformation of determinant
BD. The desired conclusion follows.
Example. n = 0, t = 2, C1 = 1, C2 = X, B = X q−1 , D = 1. Here
(7.6) α = [1, X, X q−1 , X, X q2 −1 , X, X q 3 −1 , . . .],
and α has
α = ((X q + X q−1 )α q + 1)/(X q α q + 1).
8. The approximation spectrum of elements of Class IA
Theorem 5. Suppose α is as in Theorem 4. Set sj = deg Cj (j =
1, . . . , t), st+1 = qs1, and qj = sj+1/sj (j = 1, . . . , t). Extend {qj}1≤j≤t
periodically by qj+t = qj (j = 1, 2, . . .), and put
(8.1) uj = 1 + (q − 1)/(1 + qj + qjqj+1 + . . . + qjqj+1 . . . qj+t−2)
Then the sequence {cm} introduced in (6.2) has
(8.2) lim
l→∞ cn−2+j+lt = uj (j = 1, . . . , t),
(j = 1, . . . , t).
so that S(α) consists of u1, . . . , ut (which are not necessarily distinct). Finally,
(8.3) u1 . . . ut = q.
P r o o f. Write α also as [R0, R1, . . .] and set rm = deg Rm, so that
deg Qm = r1 + . . . + rm, and by (1.12),
(8.4) cm = deg Qm+1/deg Qm = 1 + rm+1/(r1 + . . . + rm).
When
(8.5) m = n − 1 + j + lt (j = 1, . . . , t),
we have
(8.6) rm = sjq l , rm+1 = sj+1q l .
Therefore, setting u = r1 + . . . + rn−1 when n > 1, u = 0 when n = 1, and
u = −r0 when there are no terms A0, . . . , An−1, we have
r1 + . . . + rm = u + (s1 + . . . + st)(1 + q + . . . + q l−1 ) + (s1 + . . . + sj)q l
= q l ((s1 + . . . + st)/(q − 1) + s1 + . . . + sj) + O(1)
= (q l /(q − 1))(q(s1 + . . . + sj) + sj+1 + . . . + st) + O(1),