1.3. Linear patterns 57
This phenomenon extends beyond the “100% world” of exact equalities.
For any f : G C and d 1, we define the Gowers uniformity norm
f
Ud(G)
by the formula
(1.32) f
Ud(G)
:= (Eh1,...,hd,x∈GΔh1 . . . Δhd
f(x))1/2d
;
note that this is equivalent to (1.22). Using the identity
Eh,x∈GΔhf(x) =
|Ex∈Gf(x)|2
we easily verify that the expectation in the definition of (1.32) is a non-
negative real. We also have the recursive formula
(1.33) f
Ud(G)
:= (Eh∈G Δhf
2d−1
Ud−1(G)
)1/2d
for all d 1.
The U
1
norm is essentially just the mean:
(1.34) f
U 1(G)
= |Ex∈Gf(x)|.
As such, it is actually a semi-norm rather than a norm.
The U
2
norm can be computed in terms of the Fourier transform:
Exercise 1.3.18 (Fourier representation of U
2).
Define the Pontryagin dual
ˆ
G of a finite abelian group G to be the space of all homomorphisms ξ : G
R/Z. For each function f : G C, define the Fourier transform
ˆ:
f
ˆ
G C
by the formula
ˆ(ξ)
f := Ex∈Gf(x)e(−ξ(x)). Establish the identity
f
U 2(G)
=
ˆ
f
4(
ˆ)
G
:= (
ξ∈
ˆ
G
|
ˆ(ξ)|4)1/4.
f
In particular, the U
2
norm is a genuine norm (thanks to the norm prop-
erties of
4(G),
and the injectivity of the Fourier transform).
For the higher Gowers norms, there is not nearly as nice a formula known
in terms of things like the Fourier transform, and it is not immediately obvi-
ous that these are indeed norms. But this can be established by introducing
the more general Gowers inner product
(fω)ω∈{0,1}d
Ud(G)
:= Ex,h1,...,hd∈G
ω1,...,ωd∈{0,1}d
Cω1+···+ωd
fω1,...,ωd (x + ω1h1 + · · · + ωdhd)
for any
2d-tuple
(fω)ω∈{0,1}d of functions : G C, thus, in particular,
(f)ω∈{0,1}d
Ud(G)
= f
2d
Ud(G)
.
The relationship between the Gowers inner product and the Gowers unifor-
mity norm is analogous to that between a Hilbert space inner product and
Previous Page Next Page