1.2. Roth’s theorem 37
It remains to verify that Et is good. For any K 0, we have (as t is good)
that
En∈[N](1Et+1/K 1Et−1/K )
δ
1/K.
Applying Urysohn’s lemma, we can thus find a smooth function η : R
R+
with η(t ) = 1 for t t 1/K and η(t ) = 0 for t t + 1/K such that
En∈[N]|1Et (n) η(Re e(−αn + θ))|
δ
1/K.
Using the Weierstrass approximation theorem, one can then approximate
η uniformly by O(1/K) on [−1, 1] by a polynomial of degree OK(1) and
coefficients OK(1). This allows one to approximate 1Et in
L1
norm to an
accuracy of Oδ(1/K) by a function of Fourier complexity OK(1), and the
claim follows.
Corollary 1.2.9 (Correlation implies energy increment). Let f : [N]
[0, 1], and let B be a factor generated by at most M atoms, each of which
is Fourier-measurable with growth function F. Suppose that we have the
correlation
|f E(f|B),e(α·)
L2([N])
| δ
for some δ 0 and α R. Then there exists a refinement B generated
by at most 2M atoms, each of which is Fourier-measurable with a growth
function F depending only on δ, F, such that
(1.12) E(f|B )
2
L2([N])
E(f|B)
2
L2([N])
δ2.
Proof. By Lemma 1.2.8, we can find a Fourier-measurable set E with some
growth function F depending on δ, such that
|f E(f|B), 1E L2([N])| δ.
We let B be the factor generated by B and E. As 1E is measurable with
respect to B , we may project onto
L2(B
) and conclude that
|E(f|B ) E(f|B), 1E L2([N])| δ.
By Cauchy-Schwarz, we thus have
E(f|B ) E(f|B)
L2([N])
δ.
Squaring and using Pythagoras’ theorem, we obtain (1.12). The remaining
claims in the corollary follow from Exercise 1.2.10.
We can then iterate this corollary via an energy increment argument to
obtain
Proposition 1.2.10 (Weak arithmetic regularity lemma). Let f : [N]
[0, 1], and let B be a factor generated by at most M atoms, each of which
is Fourier-measurable with growth function F. Let δ 0. Then there exists
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