NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 129
Quadratic forms. Let n ≥ 3, andletn = p + q, where p ≥ 2. Let {e1,...,en} be
the standard basis for R n . If p ≥ 3, let B be the “standard" form
i
B
i=1
xiei
= 2x1xn +
p
i=2
x 2
i −
n−1
i=p+1
x 2
i . (23)
Let H = SO(B),andlet{at} be the 1-parameter subgroup of H given by ate1 = e −t e1,
atei = ei for 2 ≤ i ≤ n − 1, andaten = e t en. And let K = H ∩ ˆK, where ˆK is the
group of orthogonal matrices with determinant 1. We let dk denote the Haar measure
on K normalized so that K is a probability space.
If (p, q) = (2, 2), welet
B(x1,x2,x3,x4) = x1x4 − x2x3
be the standard form on R4 . This is the determinant on M2(R) if we identify R4 with M2(R). Note that this identification shows that SO(2, 2) is locally isomorphic
to SL2(R) × SL2(R) with the action v → g1vg −1
2 , which leaves the determinant
invariant. We let H = SL2(R) × SL2(R), K = SO(2) × SO(2), andat = (bt,bt),
where bt = diag(e −t/2 ,e t/2 ). We let dk denote the Haar measure on K normalized so
that K is a probability space. We often work with the standard lattice Z 4 and the form
Q, in which case we continue to denote by {at} and K the corresponding 1-parameter
and maximal compact subgroup of SO(Q).
If (p, q) = (2, 1), welet
B(x1,x2,x3) = x1x3 − x 2
2
be the standard form on R3 . This is the determinant on Sym2(R), the space of 2 × 2
symmetric matrices, if we identify R3 with Sym2(R). This identification shows that
SO(2, 1) is locally isomorphic to SL2(R) with the action v → gvtg, where tg is
the transpose matrix. We let H = SL2(R). We let at = diag(e−t/2 ,et/2 ),andwe
let K = SO(2) be the maximal compact subgroup of H. As before, dk denotes the
normalized Haar measure on K.
Let f be a continuous function with compact support on Rn . We define the theta
transform of f by
f ˆ(
+ ξ) =
f (v), (26)
v∈+ξ
where +ξ is any unimodular inhomogeneous lattice in R n . Note that ˆ
f is a function
on the space of inhomogeneous lattices.
(24)
(25)