Baire Category, Probabilistic Constructions and Convolution Squares

Now suppose that j(1), j(2), . . . , j(q) are distinct integers with 1 ≤ j(k) ≤ n.
By symmetry or direct calculation, the random variable
is uniformly distributed and so
q
Pr
k=1
q
k=1
mkYj(k)
mkYj(k) ∈ [−8 −1 n −q , 8 −1 n −q ]
= 4 −1 n −q .
There are no more than n q different q-tuples j(1), j(2), . . . , j(q) of the type
discussed, so, by the same kind of argument as we used in the previous
paragraph, the probability that
q
k=1
mkYj(k) ∈ [−8 −1 n −q , 8 −1 n −q ]
for any such q-tuple is no more than 1/4.
Combining the results of our last two paragraphs, we see that, provided n
is large enough, the probability that xj = Yj will fail to satisfy the conditions
of our lemma is at most 1/2. Since there must be an instance of any event
with positive probability, the required result follows.
9 Completion of the construction
The process by which we move from Lemma 8.4 to showing that H(q,p,m)
is dense looks complicated but is not. I suggest the reader concentrates on
the ideas rather than the computations.
The next exercise merely serves to establish notation.
Exercise 9.1. Let K : R → R be an infinitely differentiable function with
the following properties.
(i ′ ) K(x) ≥ 0 for all x ∈ R.
(ii ′ )
K(x)dx = 1.
R
(iii ′ ) K(x) = 0 for |x| ≥ 1/4.
If N is a positive integer and we define KN : T → R by
NK(Nt) if |t| ≤ 1/(4N),
KN(t) =
0 otherwise,
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