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

308 DUDLEY
Proof. We consider the cubes of side 2eln112 whose vertices are
of the form
t 2?Tljrl?P, 1 mj 1 < 1 + ?PUj/2r,
j-1
and the mj are integers. B is included in the union of these cubes,
their diameters are 2~, and the number of them is bounded as indi-
cated. Q.E.D.
The latter, cruder estimate in the above Lemma is sufficient for its
applications below except for one rather delicate one (Proposition
6.10).
PROPOSITION 5.7. Let C be a compact Banach ball in H and
{an> adapted to C. Then
Proof. Let s = X({a,)) and let F,, CF, C F, **a be subspaces of H,
F, n-dimensional, such that for all n, d(C, , C) < a, where
C,= CnF,.
If /3 > 01 > s then for small enough E > 0,
by (5.2). For such an E < 1 and n = n(e/2),
NC, 4 < WC, ,4).
Since r and s are homothetically invariant we can assume a, < 1.
Clearly C, is included in the block B({+}&) of Lemma 5.6, so for
E small enough
N(C, G) < exp (n (log 3 + 4 log n + log (l/e))) < exp (E-B).
Thus Y(C) 9 /3 for all B > s and r(C) < s. Q.E.D.
Proof of Proposition 5.5. By Lemma 5.0, C is compact. There is a
c > 8 such that V,(C) < n-w for n large enough. We choose a basis
(~~1 adapted to C and v, in C,,, such that e(v, , C,) = a, = d(C, C,),
n = 0, I,... . Then C includes the symmetric convex hull of the v, , so