NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

108 GUILLARMOU and TZOU
where
I1 =
M0
I2 = 2h
V (a 2 + a 2
)dvg + 2
M0
V Re ae 2iψ/h
s1
h
+ e −2iψ/h
a0 + r1
h
M0
V |a| 2 Re(e 2iψ/h )dvg, (19)
+ b0
+ b0 + a0 + s1 + r1
h
dvg. (20)
Remark 5.2
We observe from the last identity in Lemma 4.1.2 that r1/h in the expression I2 can
be replaced by the term r12 satisfying 2ir12∂ψ = b up to an error which can go in
the o(h) in (18), and similarly for the term s1/h, which can be replaced by a terms12
independent of h.
We apply the stationary phase to these two terms in Lemmas 5.3 and 5.4.
LEMMA 5.3
The estimate
I2 = 2h
M0
V Re
a(b0 + a0 +s12 +r12)dvg + o(h)
holds true where r12, s12,r12 ands12 are independent of h.
Proof
We start with the following.
LEMMA 5.4
Let f ∈ L 1 (M0). Thenash → 0,
M0
e 2iψ/h f dvg = o(1).
Proof
Since C k (M0) is dense in L 1 (M0) for all k ∈ N, it suffices to prove the lemma for
f ∈ C k (M0).Letɛ>0 be small, and choose a cutoff function χ which is identically
equal to 1 on the boundary such that
M0
χ|f | dvg ≤ ɛ.