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 ...
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SIZES OF COMPACT SUBSETS OF HILBERT SPACE 319<br />
Then Oc is a GC-set <strong>and</strong> E is not a GB-set. T/,(E) is asymptotic to a<br />
constant times<br />
<strong>and</strong> for 12 large,<br />
($yiZ n-112 (y2 n-114 5 (logj log logj)-112,<br />
Vn(Oc) > ($” (3774-1’2 fJ q(logj)-r’~,<br />
j=2<br />
Thus <strong>the</strong>re is a K > 0 such that for n large,<br />
Vn(E)/Vm(Oc) < K fi (4n2/log log j)l14,<br />
j=3<br />
which implies (b).<br />
To prove (a) it suffices to show that<br />
as E 1 0. Let S = S({a,)).<br />
Given E > 0, let<br />
fq-q(n 1% w2>>, 4/~(~(~~~~), 4 + 0<br />
N(S, c) = n = n((uj}, 6) + g rt * .<br />
Because <strong>of</strong> <strong>the</strong> slow growth <strong>of</strong> <strong>the</strong> logarithms, this implies that, for c<br />
small enough,<br />
l/98 < (log n)2 log log n < l/8,<br />
log It < l/8, log log ?z < 2 log (l/E),<br />
H(S, l ) = log ?z 2 1/5q1og (l/E))““.<br />
To estimate iV(E, E) from above we take <strong>the</strong> smallest integer 7t such<br />
that<br />
(n log .)-l12 < 42, i.e., n log n > 4/c2.<br />
For E small enough this implies n log n < 5/e2. Now<br />
where<br />
N(E, 4 < w% , 42) d WL 9 4)<br />
Em = E(W), Bn = B(GW,<br />
pi = ( j log j)-l12, j = 2 ,..., n, j!15 = 0, j > n.
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