58 1. Higher order Fourier analysis

the Hilbert space norm. In particular, we have the following analogue of the

Cauchy-Schwarz inequality:

Exercise 1.3.19 (Cauchy-Schwarz-Gowers inequality). For any tuple

(fω)ω∈{0,1}d of functions fω : G → C, use the Cauchy-Schwarz inequality

to show that

|(fω)ω∈{0,1}d

Ud(G)

| ≤

j=0,1

|(fπi,j(ω))ω∈{0,1}d

Ud(G)

|1/2

for all 1 ≤ i ≤ d, where for j = 0, 1 and ω ∈ {0,

1}d,

πi,j(ω) ∈ {0,

1}d

is formed from ω by replacing the

ith

coordinate with j. Iterate this to

conclude that

|(fω)ω∈{0,1}d Ud(G)| ≤

ω∈{0,1}d

fω Ud(G).

Then use this to conclude the monotonicity formula

f

Ud(G)

≤ f

Ud+1(G)

for all d ≥ 1, and the triangle inequality

f + g

Ud(G)

≤ f

Ud(G)

+ g

Ud(G)

for all f, g : G → C. (Hint: For the latter inequality, raise both sides to the

power

2d

and expand the left-hand side.) Conclude, in particular, that the

U

d(G)

norms are indeed norms for all d ≥ 2.

The Gowers uniformity norms can be viewed as a quantitative measure

of how well a given function behaves like a polynomial. One piece of evidence

in this direction is:

Exercise 1.3.20 (Inverse conjecture for the Gowers norm, 100% case).

Let f : G → C be such that f

L∞(G)

= 1, and let s ≥ 0. Show that

f

Us+1(G)

≤ 1, with equality if and only if f = e(φ) for some polynomial

φ: G → R/Z of degree at most s.

The problem of classifying smaller values of f

Us+1(G)

is significantly

more diﬃcult, and will be discussed in later sections.

Exercise 1.3.21 (Polynomial phase invariance). If f : G → C is a func-

tion and φ: G → R/Z is a polynomial of degree at most s, show that

e(φ)f

Us+1(G)

= f Us+1(G). Conclude, in particular, that

sup

φ

|Ex∈Ge(φ(x))f(x)| ≤ f

Us+1(G)

where φ ranges over polynomials of degree at most s.