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.