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,
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