NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 155
Now let Ɣ be a lattice in G. Then Ɣ ∩ R n is a lattice in R n . We let = ϑ(Ɣ) =
Ɣ/ Ɣ ∩ R n (this is a lattice in SLn(R)). We abuse the notation and let ϑ also denote
the projection from G/ Ɣ onto SLn(R)/. We have the following.
LEMMA A.2
Let x ∈ G/ Ɣ. Then the orbit Hϑ(x) is closed in SLn(R)/ if and only if H ⋉R n x
is closed in G/ Ɣ.
Proof
Note that closed orbits of H ⋉R n (resp. H ) have a finite H ⋉R n -invariant (resp.,
H -invariant) measure by [Mar86, Section 3]. Let x = (g, v)Ɣ. Note that H ⋉R n x =
ϑ −1 (Hϑ(x)), hence if the Hϑ(x) is closed, then so is H ⋉R n x. Suppose now that
H ⋉R n x is closed. Then Ɣ1 = H ⋉R n ∩ (g, v)Ɣ(g, v) −1 is a lattice in H ⋉R n ,
and hence Ɣ1 ∩ R n is a lattice in R n and Ɣ1/Ɣ1 ∩ R n is a lattice in H. Hence Hϑ(x)
is closed, as we wanted.
The following is special case of [EMM1, Theorem 4.4].
THEOREM A.3
Let the notation be as above. Further assume that is a lattice in H 0 ⋉R n . Let φ
be a compactly supported continuous function on H 0 ⋉R n /. Then for every ε>0
and any bounded measurable function ν on K, and for every compact subset D of
H 0 ⋉R n /, there exist finitely many points x1,...,xℓ ∈ H 0 ⋉R n / such that
(i) the orbit H 0 xi is closed and has finite H 0 -invariant measure for all i;
(ii) for any compact set F ⊂ D \
i Hxi, there exists s0 > 0 such that, for all
x ∈ F and s>s0, we have
φ(askx) ν(k) dk −
K
H 0 ⋉R n /
φdg
K
νdk
≤ ε. (89)
We also need a slight variant of [EMM1, Theorem 4.4]. This is the content of the
following.
THEOREM A.4
Let G, H, K, and Ɣ be as above. Let A ={as : s ∈ R} also be as above. Let φ be
a compactly supported continuous function on G/ Ɣ. Then for every ε>0 and any
bounded measurable function ν on K, and for every compact subset D of G/ Ɣ,there
exists finitely many points x1,...,xℓ ∈ G/ Ɣ such that
(i) the orbit H ⋉R n xi is closed and has finite H ⋉R n -invariant measure for all
i;