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.
Previous Page Next Page