Baire Category, Probabilistic Constructions and Convolution Squares

Lemma 17.7. Let Hα,n be the subset of consisting of those (E,µ) ∈ Lβ such
that we can find a finite collection of intervals I with
I ⊇ E and
|I| α+1/n < 1/n.
I∈I
I∈I
Then Hα,n is open and dense in (Lβ,dβ).
Exercise 17.8. (i) Deduce Theorem 17.6 from Lemma 17.7.
(ii) Show that, using the notation of Lemma 17.7, Hα,n is open in (Lβ,dβ).
Let us write LS,β for the set of (E,µ) ∈ Lβ with fµ infinitely differentiable.
Exercise 17.9. Show, by our usual method of convolving with a suitable Kn,
or otherwise, that, given (F,σ) ∈ Lβ and ǫ > 0, we can find an (E,µ) ∈ LS,β
with dβ (F,σ), (E,µ) ,ǫ.
Exercise 17.10. Explain why Exercises 17.8 (ii) and 17.9 enable us to reduce
the proof of Lemma 17.7 to the proof of the next lemma (Lemma 17.11).
Lemma 17.11. Let Let 1 > α > 1/2 and β = α − 1,
Given (F,σ) ∈ LS,β
2
and ǫ > 0, we can find an (E,µ) ∈ Hα,n with dβ (F,σ), (E,µ) ≤ ǫ.
Of course, Lemma 17.11 is the heart of the matter. The next two sections
are devoted to its proof.
18 More probability
The proof of Lemma 17.11 depends on the following central step.
Lemma 18.1. If 1 > γ > κ > 0 and ǫ > 0, there exist an M(α,γ) and
n0(κ,γ) ≥ 1 with the following property. If n ≥ n0, n is odd and n κ ≥ N >
n κ − 1 we can find N points
xj ∈ {r/n : r ∈ Z}
(not necessarily distinct) such that, writing
we have
and
for all 1 ≤ k ≤ n.
µ = N −1
N
j=1
δxj
|µ ∗ µ({k/n}) − n −1 | ≤ n γ−1/2
µ({k/n}) ≤ M(κ)
N
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