1.4. Equidistribution in finite fields 59
The main utility for the Gowers norms in this subject comes from the
fact that they control many other expressions of interest. Here is a basic
example:
Exercise 1.3.22. Let f : G C be a function, and for each 1 i
s + 1, let gi :
Gs+1
C be a function bounded in magnitude by 1 which
is independent of the
ith
coordinate of
Gs+1.
Let a1,...,as+1 be non-zero
integers, and suppose that the characteristic of G exceeds the magnitude of
any of the ai. Show that
|Ex1,...,xs+1∈Gf(a1x1 + · · · + as+1xs+1)
s+1
i=1
gi(x1,...,xs+1)|
f
Us+1(G)
.
(Hint: Induct on s and use (1.33) and the Cauchy-Schwarz inequality.)
This gives us an analogue of Exercise 1.3.15:
Exercise 1.3.23 (Generalised von Neumann inequality). Let Ψ=(ψ1,...,ψt)
be a collection of affine-linear forms ψi :
Gd
G with Cauchy-Schwarz com-
plexity s. If the characteristic of G is sufficiently large depending on the
linear coefficients of ψ1,...,ψt, show that one has the bound
|ΛΨ(f1,...,ft)| inf
1≤i≤t
fi
Us+1(G)
whenever f1,...,ft : G C are bounded in magnitude by one.
Conclude, in particular, that if A is a subset of G with |A| = δ|G|, then
ΛΨ(1A,..., 1A) =
δt
+ Ot( 1A δ Us+1(G)).
From the above inequality, we see that if A has some positive den-
sity δ 0 but has much fewer than
δtN d/2
(say) patterns of the form
ψ1(x),...,ψt(x) with x
Gd,
then we have
1A δ
Us+1(G) t,δ
1.
This is the initial motivation for studying inverse theorems for the Gowers
norms, which give necessary conditions for a (bounded) function to have
large U
s+1(G)
norm. This will be a focus of subsequent sections.
1.4. Equidistribution of polynomials over finite fields
In the previous sections, we have focused mostly on the equidistribution or
linear patterns on a subset of the integers Z, and in particular on intervals
[N]. The integers are of course a very important domain to study in addi-
tive combinatorics; but there are also other fundamental model examples of
domains to study. One of these is that of a vector space V over a finite field
F = Fp of prime order. Such domains are of interest in computer science
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