1.5. Inverse conjecture over finite fields 75
to the “99% inverse problem” and “1% inverse problem”. For the former,
we will soon show:
Proposition 1.5.1 (99% inverse theorem for U
d+1(V
)). Let f : V C be
such that f
L∞(V )
and f
Ud+1(V
)
1 ε for some ε 0. Then there
exists φ Poly≤d(V R/Z) such that f e(φ)
L1(V )
=
Od,F(εc),
where
c = cd 0 is a constant depending only on d.
Thus, for the Gowers norm to be almost completely saturated, one must
be very close to a polynomial. The converse assertion is easily established:
Exercise 1.5.1 (Converse to 99% inverse theorem for U
d+1(V
)). If f
L∞(V )
1 and f e(φ)
L1(V )
ε for some φ Poly≤d(V R/Z), then
F
Ud+1(V
)
1
Od,F(εc),
where c = cd 0 is a constant depending
only on d.
In the 1% world, one no longer expects to be close to a polynomial.
Instead, one expects to correlate with a polynomial. Indeed, one has
Lemma 1.5.2 (Converse to the 1% inverse theorem for U
d+1(V
)). If f : V
C and φ Poly≤d(V R/Z) are such that |f, e(φ)
L2(V
)| ε, where
f, g
L2(V
)
:= Ex∈Gf(x)g(x), then f
Ud+1(V )
ε.
Proof. From the definition (1.34) of the U
1
norm, the monotonicity of the
Gowers norms (Exercise 1.3.19), and the polynomial phase modulation in-
variance of the Gowers norms (Exercise 1.3.21), one has
|f, e(φ)| = fe(−φ)
U 1(V )
fe(−φ)
Ud+1(V
)
= f
Ud+1(V )
and the claim follows.
It is a difficult but known fact that Lemma 1.5.2 can be reversed:
Theorem 1.5.3 (1% inverse theorem for U
d+1(V
)). Suppose that char(F)
d 0. If f : V C is such that f
L∞(V )
1 and f
Ud+1(V )
ε, then
there exists φ Poly≤d(V R/Z) such that |f, e(φ)
L2(V
)|
ε,d,F
1.
This result is sometimes referred to as the inverse conjecture for the
Gowers norm (in high, but bounded, characteristic). For small d, the claim
is easy:
Exercise 1.5.2. Verify the cases d = 0, 1 of this theorem. (Hint: To verify
the d=1 case, use the Fourier-analytic identities f
U 2(V )
=(

ξ∈
ˆ
V
|
ˆ(ξ)|4)1/4
f
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