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
xα := lim
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 :
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
(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-
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] →
be a limit function. We say that x is equidistributed
with respect to a (standard) Borel probability measure μ on X if one has