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 : 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
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.
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