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

322 DUDLEY
The largest integer N in A(x) is at most (2/e)lly. Thus the number
of possible choices of A(x) is at most
N
llz < N” < exp (c(y) log (~/E)/~E~/(~+~Y))
( 1
< exp (@/(l+W))
for E small enough. For any x in C, ,
Thus
WC, ,4 G C WyW, 42)
A
where the sum is over the possible sets A = A(x) and C,(A) is the
set of all sums
Here we use a crude estimate from Lemma 5.6 to obtain for E small
enough
N(C, , E) < exp (~-~/(l+~fl)) (3&“/c)”
< exp (~-~/(l+~)).
Thus r(C,,) < 2/(1 + 201). Letting 01 t fl t y we infer
r(q) < 2/u + 2r). Q.E.D.
By Proposition 6.9, to prove Conjecture 3.3 for octahedra its uffices
to prove the following, where a, = (log n)-lj2, n > 2.
PROPOSITION 6.13. r(Oc ({am})) = 2.
Proof. Y > 2 since this Oc is not a GC-set (Corollary 3.2, Propo-
sition 6.7), or by volumes (5.8 (a) and 6.11).
To prove Y < 2 we shall use the method of the previous proof with
some additional complications. Let E > 0 and 6 > 0. Given x in Oc
let A(x) be the set of all j such that
1 Xj 1 > l 2/4Uj2 = (C” lOgj)/4* j 2 2.
Then (for E small enough) A(x) has at most 4/e2 elements. The