1.1. Equidistribution in tori 19

which roughly speaking is concerned with the limit after taking N → ∞, and

then taking a second limit by making the growth function F go to infinity.

There is a multidimensional version of Proposition 1.1.17, but we will

not describe it here; see [GrTa2011] for a statement (and also see the next

section for the ultralimit counterpart of this statement).

Remark 1.1.21. These single-scale abelian equidistribution theorems are a

special case of a more general single-scale nilpotent equidistribution theorem,

which will play an important role in later aspects of the theory, and which

was the main result of the aforementioned paper of Ben Green and myself.

As an example of this theorem in action, we give a single-scale strength-

ening of Exercise 1.1.8 (and Exercise 1.1.15):

Exercise 1.1.23 (Recurrence). Let P be a polynomial map from Z to

Td

of degree s, and let N ≥ 1 be an integer. Show that for every ε 0 and

N 1, and every integer n0 ∈ [N], we have

|{n ∈ [N] : P (n) − P (n0) ≤ ε}|

d,s

εOd,s(1)N.

Exercise 1.1.24 (Multiple recurrence). With the notation of Exercise 1.1.23,

establish that

|{r ∈[−N, N] : P (n0 + jr) − P (n0) ≤ ε for j = 0, 1,...,k − 1}|

d,s,k

εOd,s,k(1)N

for any k ≥ 1.

Exercise 1.1.25 (Syndeticity). A set of integers is syndetic if it has bounded

gaps (or equivalently, if a finite number of translates of this set can cover

all of Z). Let P : Z →

Td

be a polynomial and let ε 0. Show that

the set {n ∈ Z: P (n) − P (n0) ≤ ε} is syndetic. (Hint: First reduce to

the case when P is (totally) asymptotically equidistributed. Then, if N is

large enough, show (by inspection of the proof of Exercise 1.1.21) that the

translates P (· + n0) are ε-equidistributed on [N] uniformly for all n ∈ Z,

for any fixed ε 0. Note how the asymptotic theory and the single-scale

theory need to work together to obtain this result.)

1.1.3. Ultralimit equidistribution theory. The single-scale theory was

somewhat more complicated than the asymptotic theory, in part because

one had to juggle parameters such as N, δ, and (for the equidistribution

theorems) F as well. However, one can clean up this theory somewhat

(especially if one does not wish to quantify the dependence of bounds on

the equidistribution parameter δ) by using an ultralimit, which causes the δ