1.7. Linear equations in primes 115

and the claim follows.

Exercise 1.7.2. Show that the estimate (1.54) for q ≤ 2 can only hold when

ν is bounded uniformly in N; this explains the presence of the hypothesis

q 2 in the above condition.

Exercise 1.7.3. Show that the estimate (1.54) is equivalent to the estimate

En∈Z/NZ|

ξ∈Z/NZ

g(ξ)e(ξnx/N)|ν(n) ≤ C g

q (Z/NZ)

for all g : Z/NZ → C, where q := q/(q − 1) is the dual exponent to q.

Informally, this asserts that a Fourier series with

q

coeﬃcients can be “re-

stricted” to the support of ν in a uniformly absolutely integrable manner

(relative to ν). Historically, this is the origin of the term “restriction theo-

rem” (in the context where Z/NZ is replaced with a Euclidean space such

as

Rn,

and ν is a surface measure on a manifold such as the sphere

Sn−1).

See for instance [Ta2003].

Now we turn to the approximation property. The approximation g to

f needs to be close in

u2

norm, i.e., the Fourier coeﬃcients need to be

uniformly close. One attempt to accomplish this is hard thresholding: one

simply discards all Fourier coeﬃcients in the Fourier expansion

f(n) =

ξ∈Z/NZ

ˆ(ξ)e(xξ/N)

f

of f that are too small, thus setting g equal to something like

g(n) =

ξ∈Z/NZ:|

ˆ(ξ)|≥ε

f

ˆ(ξ)e(xξ/N).

f

The main problem with this choice is that there is no guarantee that the

non-negativity of f will transfer over to the non-negativity of g; also, there

is no particular reason why g would be bounded.

But a small modification of this idea does work, as follows. Let S :=

{ξ ∈ Z/NZ : |

ˆ(ξ)|

f ≥ ε} denote the large Fourier coeﬃcients of f. The

function g proposed above can be viewed as a convolution f ∗ K, where

K(n) :=

∑

ξ∈S

e(xξ/N) and f ∗ K(n) := Em∈Z/NZf(m)K(n − m). The

inability to get good pointwise bounds on f ∗ K can be traced back to the

oscillatory nature of the convolution kernel K (which can be viewed as a

generalised Dirichlet kernel).

But experience with Fourier analysis tells us that the behaviour of such

convolutions improves if one replaces the Dirichlet-type kernels with some-

thing more like a Fej´ er-type kernel instead. With that in mind, we try

g(n) := Em1,m2∈Bf(n + m1 − m2)