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