NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 133
above. Define
f ˜(g
: qξ) =
v∈X(qξ )
f (gv). (36)
The following is an analogue of [EMM2, Theorem 2.3] and will provide us with
the upper bound required for the proof of Theorem 1.9. The proof of this theorem is
the main technical part of this paper and will occupy the rest of this paper.
THEOREM 4.1
Let G, H, K, and {at} be as in Section 2 for the signature (2, 2) case. Let Qξ be a
quadratic form of signature (2, 2) which is Diophantine. Let q ∈ SL4(R) and qξ be
as above. Let ν be a continuous function on K. Then we have
lim sup
t→∞
˜f (atk : qξ)ν(k) dk ≤ f ˆ(g)
dµ(g)
K
G/ Ɣ
K
ν(k) dk, (37)
where µ is the G-invariant probability measure on G/ Ɣ if the homogenous part, Q,
is irrational and where the H ⋉R 4 -invariant probability measure on the closed orbit
is H ⋉R 4 · qξ if Q is a rational form.
Proof of Theorem 1.9
Suppose that Qξ is as in the statement of Theorem 1.9. An argument like that
of [EMM1, Sections 3.4, 3.5] combined with Theorem 4.1 gives: if 0 /∈ (a,b),
then
lim sup
T →∞
NQ,ξ,(a,b,T ) = lim sup ÑQ,ξ,(a,b,T ) ≤ λQ,(b − a)T
T →∞
2 . (38)
This upper bound, combined with the lower bound obtained by Theorem 1.4, proves
Theorem 1.9.
The proof of Theorem 4.1 extensively utilizes results and ideas from [EMM1]
and [EMM2], and we will try to use their terminology and notation for the convenience
of the reader. We recall these theorems and terminology when we need them. Let us
start with the following.
THEOREM 4.2
Let {at} and K be as in Theorem 4.1. Let be any lattice in R 4 . Then for i = 1, 3
and any ε>0, we have
sup αi(atk)
t>0 K
2−ε dk < ∞. (39)