8 1. Higher order Fourier analysis
Applying Cauchy-Schwarz, we conclude
En∈[N]an
(En∈[N]|Eh∈[H]an+h|2)1/2
+
H
N
.
We expand out the left-hand side as
En∈[N]an (Eh,h
∈[H]
En∈[N]an+han+h
)1/2
+
H
N
.
The diagonal contribution h = h is O(1/H). By symmetry, the off-diagonal
contribution can be dominated by the contribution when h h . Making
the change of variables n n h , h h + h (accepting a further error of
O(H1/2/N 1/2)),
we obtain the claim.
Corollary 1.1.7 (van der Corput lemma). Let x: N
Td
be such that the
derivative sequence ∂hx: n x(n+h)−x(n) is asymptotically equidistributed
on N for all positive integers h. Then xn is asymptotically equidistributed
on N. Similarly with N replaced by Z.
Proof. We just prove the claim for N, as the claim for Z is analogous (and
can in any case be deduced from the N case).
By Proposition 1.1.2, we need to show that for each non-zero k
Zd,
the exponential sum
|En∈[N]e(k · x(n))|
goes to zero as N ∞. Fix an H 0. By Lemma 1.1.6, this expression is
bounded by
(Eh∈[H]|En∈[N]e(k · (x(n + h)
x(n)))|)1/2
+
1
H1/2
+
H1/2
N 1/2
.
On the other hand, for each fixed positive integer h, we have from hypothesis
and Proposition 1.1.2 that |En∈[N]e(k · (x(n + h) x(n)))| goes to zero as
N ∞. Taking limit superior as N ∞, we conclude that
lim sup
N→∞
|En∈[N]e(k · x(n))|
1
H1/2
.
Since H is arbitrary, the claim follows.
Remark 1.1.8. There is another famous lemma by van der Corput con-
cerning oscillatory integrals, but it is not directly related to the material
discussed here.
Corollary 1.1.7 has the following immediate corollary:
Corollary 1.1.9 (Weyl equidistribution theorem for polynomials). Let s
1 be an integer, and let P (n) =
αsns
+ · · · + α0 be a polynomial of degree s
with α0,...,αs
Td.
If αs is irrational, then n P (n) is asymptotically
equidistributed on Z.
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