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

320 DUDLEY
By Lemma 5.6, for E small and hence for n large enough,
N(& ) E/2) < fi (2 + nl’2( j log$l’“/~)
j=2
(Note: the logarithms have served to make n smaller, but they
are no longer needed.)
For n large we have n! > (n/e)“, so
N(E, l ) < (3e/0z112)” = exp {n[log 3 + 1 + log (l/c) - 3 log n]}.
Since n < 5/e2, we have, for E small enough,
log n < log 5 + 2 log (l/c) < 3 log (l/c),
n > 4/e2 log n > 4/3c2 log (l/c),
log 12 > log (4/3) + 2 log (l/E) - log log (l/E),
H(E, 6) < 5[3 + log log (1 /c)]/e” log (1 /e).
Thus H(E, e)/H(S, e) -+ 0 as E J 0. Q.E.D.
Suppose given a sufficient condition that a set C be a GC-set,
asserting that H(C, 6) is sufficiently small (e.g., Theorem 2.1) or
that the V,(C) are sufficiently small (e.g., Proposition 5.5, Conjecture
5.4). Then the GC-octahedron of Proposition 6.10 will never satisfy
such a condition since the ellipsoid does not. Hence no such
sufficient conditon can be necessary.
In the converse direction, likewise, a sufficient condition for a Banach
ball not to be a GB-set such as Theorem 5.3 or Conjecture 3.3 cannot
be necessary.
One may, however, seek “best possible” conditions of the given
kinds. In the four cases, Theorem 3.1 has a fairly strong claim to be
best (see the next section). Theorem 5.3 has a weaker claim. Conjec-
tures 3.3 and 5.4, if they are true, could probably be improved upon.
The volume of the n-dimensional octahedron
is 2”/n!, which is asymptotic to (2e)“/n”(2m)1/2 by Stirling’s formula.
Thus by 5.11 and 5.12