NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 131
Theorem 1.6 is proved using the following, which is a result of combining Theorems
3.1, A.3, andA.4. We have the following.
THEOREM 3.2 ([EMM1, Theorem 3.5])
Suppose that p ≥ 3 and q ≥ 1. Let f and fˆ be as above. Let ν be any continuous
function on K. Then, for every compact subset D of G/ Ɣ, there exist finitely many
points x1,...,xℓ ∈ G/ Ɣ such that
(i)
(ii)
the orbit Hxi is closed and has finite H -invariant measure for all i;
for any compact set F ⊂ D \
i Hxi, there exists t0 > 0 such that, for all
x ∈ F and t>t0, we have
f ˆ(atkx)
ν(k) dk −
fdµ ˆ
νdk
≤ ε, (31)
K
G/ Ɣ K
where either µ is the G-invariant measure on G/ Ɣ or H ⋉R n x is closed and has
H ⋉R n -invariant probability measure and µ is this measure.
Proof
We may and will assume that φ is nonnegative. Now define
A(r) = ∈ G/ Ɣ : α1() >r . (32)
Let gr be a continuous function on G/ Ɣ such that gr() = 0 if /∈ A(r),gr() = 1
for all ∈ A(r + 1), and0≤ gr() ≤ 1 if r ≤ α1() ≤ r + 1. We have
f ˆ = ( fˆ − fgr) ˆ + fgr. ˆ Note that fˆ − fgr ˆ is a continuous function with compact
support on G/ Ɣ.
Note that H 0 ⋉Rn is a maximal connected subgroup of G. Hence for every
δ>0, there exists r0 such that if H ⋉Rny is a closed orbit of H ⋉Rn in G/ Ɣ with
an H ⋉Rn-invariant probability measure σ ,thenσ (A(r) ∩ H ⋉Rny) r0. Hence for r sufficiently large, we get
fdµ− ˆ ( fˆ − fgr) ˆ
dµ