18 1. Higher order Fourier analysis

Proof. The case s = 0 is trivial, so suppose inductively that s ≥ 1, and that

the claim has already been proven for lower degrees. Then for fixed degree,

the case d = 0 is vacuously true, so we make a further inductive assumption

d ≥ 1 and the claim has already been proven for smaller dimensions (keeping

s fixed).

If P is already totally 1/F (1)-equidistributed then we are done (setting

Pequi = P and Psmth = Prat = 0 and M = 1), so suppose that this is not

the case. Applying Exercise 1.1.21, we conclude that there is some non-zero

k ∈

Zd

with |k|

d,s

F

(1)Od,s(1)

such that

k · αi

T d,s

F

(1)Od,s(1)/N i

for all i = 0,...,s. We split k = mk where k is irreducible and m is a posi-

tive integer. We can therefore split αi = αi,smth + αi,rat + αi where αi,smth =

O(F

(1)Od,s(1)/N i),

qαi = 0 for some positive integer q = Od,s(F

(1)Od,s(1)),

and k · αi = 0. This then gives a decomposition P = Psmth + P + Prat,

with P taking values in the subtorus {x ∈

Td

: k · x = 0}, which can be

identified with

Td−1

after an invertible linear transformation with integer

coeﬃcients of size Od,s(F

(1)Od,s(1)).

If one applies the induction hypothesis

to P (with F replaced by a suitably larger function F ) one then obtains

the claim.

The final claim about polynomial bounds can be verified by a closer

inspection of the argument (noting that all intermediate steps are polyno-

mially quantitative, and that the length of the induction is bounded by

Od,s(1)).

Remark 1.1.20. It is instructive to see how this smooth-equidistributed-

rational decomposition evolves as N increases. Roughly speaking, the torus

T that the Pequi component is equidistributed on is stable at most scales,

but there will be a finite number of times in which a “growth spurt” oc-

curs and T jumps up in dimension. For instance, consider the linear flow

P (n) := (n/N0,n/N0

2)

mod

Z2

on the two-dimensional torus. At scales

N N0 (and with F fixed, and N0 assumed to be suﬃciently large de-

pending on F ), P consists entirely of the smooth component. But as N

increases past N0, the first component of P no longer qualifies as smooth,

and becomes equidistributed instead; thus in the range N0 N N0

2,

we

have Psmth(n) = (0,n/N0

2)

mod

Z2

and Pequi(n) = (n/N0, 0) mod

Z2

(with

Prat remaining trivial), with the torus T increasing from the trivial torus

{0}2

to

T1

× {0}. A second transition occurs when N exceeds N0

2,

at which

point Pequi encompasses all of P . Evolving things in a somewhat different

direction, if one then increases F so that F (1) is much larger than N0

2,

then

P will now entirely consist of a rational component Prat. These sorts of

dynamics are not directly seen if one only looks at the asymptotic theory,