On the Characters and the Plancherel Formula of Nilpotent Groups ...

318 DUDLEY
PROPOSITION 6.9. The following are equivalent:
(a) Oc ({b,}) is a GB-set;
(b) S({b,}) is a GB-set;
(c) b, = O((log $-l/2).
Proof. We use some notation and results of the previous proof.
(By the way, note that Theorem 4.7 and either of 6.7 and 6.9 make
the other at least very plausible.) Here the equivalence of (a) and (b)
is obvious. (b) is equivalent to the statement that for some M > 0,
4t I GJTJ I < M f or n sufficiently large, with probability 1, or that
(6.8) converges for E = M. If (c) holds, i.e., if for some N > 0,
1 ollz 1 < N for all n, we can let M = 2N and infer (b). If {OIJ is
unbounded, then given M we choose n so that 01, > 2M. Then
01~ > M for n1i2 < j < n,
C
d’= 0. Thus (b) implies (c). Q.E.D.
We infer from Propositions 6.3 and 6.7 that a GC-set, Oc ({l/log n)),
is not included in any GB-ellipsoid, since
F2 u(logn)2 = + cc
(see [1.5], Lemma 2).
We next show that the GC- and GB-properties are not monotone
functions of the “size” of a set as measured by volumes V, or by
e-entropy.
PROPOSITION 6.10. There exist a GC-set Oc = Oc ({a,}) and u
non-GB-ellipsoid E = E({b,J) such that
(a) H(E, e)/H(Oc, c) --+ 0 as c: JO,
(b) V,(E)/&(Oc) -+ 0 as n + 00.
Proof. We let a, = an(log n)-l12, n > 2, where 01~ J 0 sufficiently
slowly; for definiteness we can let 0~~ = (log log n)-l14, n > 3. Let
b, = (n log n log log ?z)-1’2, n > 3.