1.1. Equidistribution in tori 21

Given a sequence of functions fα : Xα → Yα, we can form the ultralimit

limα→α∞ fα : limα→α∞ Xα → limα→α∞ Yα by the formula

( lim

α→α∞

fα) lim

α→α∞

xα := lim

α→α∞

fα(xα);

one easily verifies that this is a well-defined function between the two ultra-

products. We refer to ultralimits of functions as limit functions; they are

roughly analogous to “simple functions” in measurable theory. We identify

every standard function f : X → Y with its ultralimit limα→α∞ f :

∗X

→

∗Y

,

which extends the original function f.

Now we introduce limit asymptotic notation, which is deliberately chosen

to be similar (though not identical) to ordinary asymptotic notation. Given

two limit numbers X, Y , we write X Y , Y X, or X = O(Y ) if we

have |X| ≤ CY for some standard C 0. We also write X = o(Y ) if we

have |X| ≤ cY for every standard c 0; thus for any limit numbers X, Y

with Y 0, exactly one of |X| Y and X = o(Y ) is true. A limit real

is said to be bounded if it is of the form O(1), and infinitesimal if it is of

the form o(1); similarly for limit complex numbers. Note that the bounded

limit reals are a subring of the limit reals, and the infinitesimal limit reals

are an ideal of the bounded limit reals.

Exercise 1.1.26 (Relation between limit asymptotic notation and ordinary

asymptotic notation). Let X = limα→α∞ Xα and Y = limα→α∞ Yα be two

limit numbers.

(i) Show that X Y if and only if there exists a standard C 0 such

that |Xα| ≤ CYα for all α suﬃciently close to α0.

(ii) Show that X = o(Y ) if and only if, for every standard ε 0, one

has |Xα| ≤ εYα for all α suﬃciently close to α0.

Exercise 1.1.27. Show that every bounded limit real number x has a

unique decomposition x = st(x) + (x − st(x)), where st(x) is a standard

real (called the standard part of x) and x − st(x) is infinitesimal.

We now give the analogue of single-scale equidistribution in the ultra-

limit setting.

Definition 1.1.23 (Ultralimit equidistribution). Let X = (X, d) be a stan-

dard compact metric space, let N be an unbounded limit natural number,

and let x: [N] →

∗X

be a limit function. We say that x is equidistributed

with respect to a (standard) Borel probability measure μ on X if one has

stEn∈[N]f(x(n)) =

X

f dμ