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

Thus
where
SIZES OF COMPACT SUBSETS OF HILBERT SPACE 311
Thus the following is useful.
PROPOSITION 5.12. If b, JO,
-WV(C)) = EW(C) + 4%)) (5.11)
ew((bJ) = li%tup (i log bj)/n log n.
j=l
eea(fbj,)) = - l/x({bjl)*
Proof. Given 8 > 0, we have by (5.2):
n(c) = n({b,}, e) < l/CA+8
for E small enough, and n(e) > l/&-B for arbitrarily small E > 0. Now
if 71 = n(e),
(~lw,)/~logn >(hsM%4.
When n >, l/~~+ and 0 < E < 1,
log n 2 (A - 8) log (1 /C) and (log c)/log n > - l/(h - 6).
Thus letting 6 J 0 we have
e@d) >, - l/X.
For the converse inequality, we can assume b, < 1. For any positive
integer m let E = e(m) satisfy m = E-X-~~. Then as m -F CO, E JO.
Since
n(c) < l/E”+8 < l/&+26
for E small enough,
(fJl log b,)/m log m < (m - n(e(m))) (log 6) th+28/(h + 28) log (l/c)
< (1 - 4 (log 4/(X + 26) log u/4
= (- 1 + @)/(A + 26) + - l/(A + 26),
where E = e(m), m -P co. Thus, letting 6 JO, we have
~(VJ,)) < - l/h.
Q.E.D.