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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 317
Now (b) is equivalent by the zero-one law ([13] A, p. 228) to the
following: for any E > 0,
or
i L
n-1 I
eox2J2 dx < co.
00
e-x2/2 dx is asymptotic to e-M2/2/M.
As is well known, an integration by parts shows that as M --+ co,
s M
Thus (b) is equivalent to
5 b, exp (- c2/2bn2) < co.
n-1
Letting b, = olyr (log n)-l12, n > 2, we obtain the series
f c$(log 4-l/2 d’~,a. (63)
n=2
If (c) holds, i.e., ollt --t 0, then the terms of (6.8) become less than
W-~ for n large, so (b) holds.
Conversely suppose (c) is false, so that for some 6 > 0, ol, > 6
for arbitrarily large values of 7t. For such an n and nil2 42.
>, & - ?.w - 1)/2?21’2 log n + co
as n + co (recall that 6 is independent of n). Thus (6.8) diverges and
(b) fails, so (b) 3 (c). Q.E.D.