NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 109
Then, splitting the integral and using stationary phase for the 1 − χ term, we obtain
e 2iψ/h
f dvg
≤ (1 − χ)e 2iψ/h
f dvg
+ χe 2iψ/h
f dvg
≤ ɛ + Oɛ(h),
M0
M0
which concludes the proof by taking h small enough depending on ɛ.
The proof of Lemma 5.3 is a direct consequence of Lemma 5.4 and Remark
5.2.1.
The second lemma will be proved at the end of this section.
LEMMA 5.5
The estimate
I1 =
M0
V (a 2 + a 2 )dvg + hCpV (p)|a(p)| 2 Re(e 2iψ(p)/h ) + o(h)
holds true with Cp = 0 and is independent of h.
With Lemmas 5.3 and 5.4, we can write (18) as
0 = V (a 2 + a 2 )dvg + O(h)
M0
and thus we can conclude that
0 =
M0
M0
V (a 2 + a 2 )dvg.
Therefore, (18) becomes, with Lemma 5.3,
0 = CpV (p)|a(p)| 2 Re(e 2iψ(p)/h
) + 2 V Re a(b0 + ˜s12 + a0 + ˜r12) dvg + o(1).
M0
Since ψ(p) = 0, we may choose a sequence of hj → 0 such that Re(e 2iψ(p)/hj ) = 1
and another sequence ˜hj → 0 such that Re(e 2iψ(p)/˜hj ) =−1 for all j. Adding the
expansion with h = hj and h = ˜hj, we deduce that
0 = 2CpV (p)|a(p)| 2 + o(1)
as j →∞, and since Cp = 0, a(p) = 0, we conclude that V (p) = 0. The set
of p ∈ M0 for which we can conclude this is dense in M0, by Proposition 2.3.1.
Therefore, we can conclude that V (p) = 0 for all p ∈ M0.
We now prove Lemma 5.5.