64 1. Higher order Fourier analysis
Exercise 1.4.9. Let P : V → F be a limit classical polynomial which is
linear (i.e., of degree at most 1). Show that P is equidistributed if and only
if P is non-constant.
There is an analogue of the Weyl equidistribution criterion in this setting.
Call a limit function P : V → F biased if |Ex∈V e(P (x))| 1, and unbiased
if Ex∈V e(P (x)) = o(1), where we identify P (x) ∈ F with an element of R/Z.
Exercise 1.4.10 (Weyl equidistribution criterion). Let P : V → F be a
limit function. Show that P is equidistributed if and only if kP is unbiased
for all non-zero k ∈ F.
Thus to understand the equidistribution of polynomials, it suﬃces to un-
derstand the size of exponential sums Ex∈V e(P (x)). For linear polynomials,
this is an easy application of Fourier analysis:
Exercise 1.4.11. Let P : V → F be a polynomial of degree at most 1.
Show that |Ex∈V e(P (x))| equals 1 if P is constant, and equals 0 if P is not
constant. (Note that this is completely consistent with the previous two
Next, we turn our attention to the quadratic case. Here, we can use the
Weyl differencing trick, which we phrase as an identity
= Eh∈V Ex∈V Δhf(x)
for any finite vector space V and function f : V → C, where Δhf(x) :=
f(x+h)f(x) is the multiplicative derivative. Taking ultralimits, we see that
the identity also holds for limit functions on limit finite dimensional vector
spaces. In particular, we have
(1.36) |Ex∈V e(P
= Eh∈V Ex∈V e(∂hP (x))
for any limit function P : V → F on a limit finite dimensional space.
If P is quadratic, then ∂hP is linear. Applying (1.4.11), we conclude
that if P is biased, then ∂hP must be constant for |V | values of h ∈ V .
On the other hand, by using the cocycle identity
∂h+kP (x) = ∂hP (x + k) + ∂kP (x)
we see that the set of h ∈ V for which ∂hP is constant is a limit subspace
of W . On that subspace, P is then linear; passing to a codimension one
subspace W of W , P is then constant on W . As ∂hP is linear for every
h, P is then linear on each coset h + W of W . As |W | |V |, there are
only a bounded number of such cosets; thus P is piecewise linear, and thus
piecewise constant on slightly smaller cosets. Intersecting all the subspaces
together, we can thus find another limit subspace U with |U| |V | such