1.7. Linear equations in primes 109

[GrTa2011] for details. We give a special case of Theorem 1.6.16 as an

exercise:

Exercise 1.6.22. Use Lemma 1.6.15 to show that if α, β are two real num-

bers that are linearly independent modulo 1 over the integers, then the

polynomial orbit

n →

⎛

⎝0

1 αn 0

1

βn⎠

0 0 1

⎞

Γ

is asymptotically equidistributed in the Heisenberg nilmanifold G/Γ; note

that this is a special case of Theorem 1.6.16. Conclude that the map n →

αn βn mod 1 is asymptotically equidistributed in the unit circle.

One application of this equidistribution theory is to show that bracket

polynomial objects such as (1.52) have a negligible correlation with any gen-

uinely quadratic phase n →

e(αn2

+ βn + γ) (or more generally, with any

genuinely polynomial phase of bounded degree); this result was first estab-

lished in [Ha1993]. On the other hand, from Theorem 1.6.12 we know that

(1.52) has a large U 3[N] norm. This shows that even when s = 2, one cannot

invert the Gowers norm purely using polynomial phases. This observation

first appeared in [Go1998] (with a related observation in [FuWi1996]).

Exercise 1.6.23. Let the notation be as in Exercise 1.6.22. Show that

lim

n→∞

En∈[N]e(αn βn −

γn2

− δn) = 0

for any γ, δ ∈ R. (Hint: You can either apply Theorem 1.6.16, or go back

to Lemma 1.6.15.)

1.7. Linear equations in primes

In this section, we discuss one of the motivating applications of the theory

developed thus far, namely to count solutions to linear equations in primes

P = {2, 3, 5, 7,... } (or in dense subsets A of primes P). Unfortunately, the

most famous linear equations in primes, the twin prime equation p2 −p1 = 2

and the even Goldbach equation p1 + p2 = N, remain out of reach of this

technology (because the relevant aﬃne linear forms involved are commensu-

rate, and thus have infinite complexity with respect to the Gowers norms),

but most other systems of equations, in particular, that of arithmetic pro-

gressions pi = n + ir for i = 0,...,k − 1 (or equivalently, pi + pi+2 = 2pi+1

for i = 0,...,k −2) , as well as the odd Goldbach equation p1 +p2 +p3 = N,

are tractable.