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

exercises.)

Next, we turn our attention to the quadratic case. Here, we can use the

Weyl differencing trick, which we phrase as an identity

(1.35) |Ex∈V

f(x)|2

= 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

(x))|2

= 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