1.1. Equidistribution in tori 3

The theory of equidistribution of polynomial orbits was developed in

the linear case by Dirichlet and Kronecker, and in the polynomial case by

Weyl. There are two regimes of interest; the (qualitative) asymptotic regime

in which the scale parameter N is sent to infinity, and the (quantitative)

single-scale regime in which N is kept fixed (but large). Traditionally, it is

the asymptotic regime which is studied, which connects the subject to other

asymptotic fields of mathematics, such as dynamical systems and ergodic

theory. However, for many applications (such as the study of the primes), it

is the single-scale regime which is of greater importance. The two regimes

are not directly equivalent, but are closely related: the single-scale theory

can be usually used to derive analogous results in the asymptotic regime,

and conversely the arguments in the asymptotic regime can serve as a sim-

plified model to show the way to proceed in the single-scale regime. The

analogy between the two can be made tighter by introducing the (qualita-

tive) ultralimit regime, which is formally equivalent to the single-scale regime

(except for the fact that explicitly quantitative bounds are abandoned in the

ultralimit), but resembles the asymptotic regime quite closely.

For the finitary portion of the text, we will be using asymptotic notation:

X Y , Y X, or X = O(Y ) denotes the bound |X| ≤ CY for some

absolute constant C, and if we need C to depend on additional parameters,

then we will indicate this by subscripts, e.g., X

d

Y means that |X| ≤ CdY

for some Cd depending only on d. In the ultralimit theory we will use an

analogue of asymptotic notation, which we will review later in this section.

1.1.1. Asymptotic equidistribution theory. Before we look at the

single-scale equidistribution theory (both in its finitary form, and its ultra-

limit form), we will first study the slightly simpler, and much more classical,

asymptotic equidistribution theory.

Suppose we have a sequence of points x(1),x(2),x(3),... in a compact

metric space X. For any finite N 0, we can define the probability measure

μN := En∈[N]δx(n)

which is the average of the Dirac point masses on each of the points x(1),...,

x(N), where we use En∈[N] as shorthand for

1

N

∑N

n=1

(with [N] :=

{1,...,N}). Asymptotic equidistribution theory is concerned with the lim-

iting behaviour of these probability measures μN in the limit N → ∞, for

various sequences x(1),x(2),... of interest. In particular, we say that the

sequence x: N → X is asymptotically equidistributed on N with respect to a

reference Borel probability measure μ on X if the μN converge in the vague

topology to μ or, in other words, that

(1.1) En∈[N]f(x(n)) =

X

f dμN →

X

f dμ