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

328 DUDLEY
formly in t with probability 1, so xt is sample-continuous. Then if X
is the set of all x1 in H, r(X) < 2.
Pyoof. CMIt,I + hlbndh ence (1”) converge uniformly in t
by the three-series theorem. We represent X in H as follows: let
{qn} be an orthonormal basis, and
X = I f &J{cos nt) y2% + (sin nt) y+i] : 0 4 t < 27f
t .
n=1
Then X C B({b,)) where /I, = bznmI = b,, , and C b, < a~. As
remarked after Proposition 6.6, r(B) < 2, SO r(X) < 2. Q.E.D.
For “lacunary” random Fourier series of the form
%(a) = fj Igk!s&J) cos bk%
k=l
where nkfl/nk > y > 1 for all k, & > 0 and the & are independent
normalized Gaussian random variables, it is easy to see that C t, < 00
implies C & < co. Thus Kahane’s theorem and Proposition 7.3
together imply that Conjecture 3.3 holds for lacunary series (without
any further monotonicity assumptions).
8. COMMENTS ON THE CONJECTURES
Of the three Conjectures 3.3,5.4, and 5.9, Conjecture 5.4 is supported
by 5.9 which has nothing a priori to do with Gaussian processes.
One might seek similar support for 3.3. But Y(C) > 2 does not imply
(for octahedra) that the W,(C) are too large to satisfy Theorem 5.3.
However, Theorem 5.3 may not be the best possible, and the largest
V,(C) and W,(C) I know for a GB-set C are those of Oc({K(log n)-‘j2}),
K > 0, namely
VJC) 2 W,(C) = $ fi (logj)-1’2.
* 3=2
The following approach to some of our problems might seem
natural, Given E > 0 and C, C Rn, let C,c be the set of points within
E of C, . Then
N(G ,24 B L.(C,“)lW.
For C, convex, X,(C,S) can be expressed in terms of “mixed volumes”