In 1955
Cassels and
Swinnerton-Dyer showed that Littlewood's Conjecture would follow from the following conjecture in the
geometry of numbers in the case n=3:
Conjecture 1: Let
L be the product of
n linear forms on \mathbb{R}^n. Suppose n\geq 3 and
L is not a multiple of a form with integer coefficients. Then \inf\{|L(x)|\mid x\in\mathbb{Z}^n\setminus\{0\}\}=0. Conjecture 1 is equivalent to the following conjecture concerning the orbits of the diagonal subgroup
D on SL(n, \mathbb{R})/SL(n, \mathbb{Z}), as was essentially noticed by Cassels and Swinnerton-Dyer.
Conjecture 2: Let n\geq 3. For any x\in SL(n, \mathbb{R})/SL(n, \mathbb{Z}), if the orbit Dx is relatively compact, then Dx is closed. This is due to
Margulis. Conjecture 2 is a special case of the following far more general conjecture, also due to Margulis.
Conjecture 3: Let
G be a connected Lie group, \Gamma a lattice in
G, and
H a closed connected subgroup generated by (Ad_G, \mathbb{R})-split elements, i.e. all eigenvalues of Ad_G(g) are real for each generator
g. Then for any x\in G/\Gamma, exactly one of the following holds: • \overline{Hx} is homogeneous, i.e. there is a closed subgroup
F of
G such that \overline{Hx}=Fx. • There exists a closed connected subgroup
F of
G and a continuous epimorphism \phi from
F onto a Lie group
L such that H\subset F, Fx is closed in G/\Gamma, \phi(F_x) is closed in
L where F_x is the stabilizer, and \phi(H) is a one-parameter subgroup of
L containing no non-trivial Ad_L-unipotent elements, i.e. elements
g for which 1 is the only eigenvalue of Ad_L(g). ==Partial results==