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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 321
A sequence b, J- 0 such that A({&}) < co is o((log n)-li2) (cf. end
of the proof of Proposition 6.7). Thus conjecture 5.4 holds for octa-
hedra. The next proposition implies that conjecture 5.9 also holds for
octahedra.
PROPOSITION 6.12. Let X = A({&}), s = EV(Oc ({b,))), I = t(Oc).
Then r = - 2/(2s + 1) = 2h/(2 + X) ;f any of these terms is less than
2 (i.e., if r < 2, s < - 1, or X < co). Thus under these conditions
Oc is volumetric.
Proof. r > - 2/p + 1) in general by 5.8 (a). If s < 1, then
X < co and s > - 1 - l/h by 6.11. Thus any of the hypotheses
implies X < co, and then l/X >, - 1 - s,
W(2 + 4 = 2/((2/4 + 1) < - 2/p + 1)
ifs < - 4. It will now suffice to show that if h < co,
r < 2/((2/4 + 1)
(since then Y < 2 and s < - 1 < - +).
Let 0 < y < l/X. Then for n large enough, b, < l/n7 by 5.2.
Thus for some K > 0, Oc ({b,}) C KC, where C, = Oc ((l@)),
and r(Oc) < r(KC,) = r(C,). Thus it is enough to prove that
+q < 2/u + &).
For x in C, , we have
x = 1 xjP)jl!P, Cbjl 0, let A(x) be the set of all j such that
1 x, 1 > j2Q2/4.
Then the number m of integers in A(x) satisfies
m1+2y/(l + 2~) = Sr x2Y dx < ,gi jay < 41~~.
Let (II < /3 < y. Then for some c(r),
m < c(.y)/&U+2v) < &/U+W)
for E small enough. (Of course m depends on y and E).