Littlewood conjecture
Template:Short description In mathematics, the Littlewood conjecture is an open problem in Diophantine approximation, proposed by J. E. Littlewood around 1930. It states that for any two real numbers <math>\alpha</math> and <math>\beta</math>,
- <math>\liminf_{n\to\infty} \ n\,\Vert n\alpha\Vert \,\Vert n\beta\Vert = 0,</math>
where <math>\Vert x\Vert=\min(|x-\lfloor x \rfloor|,|x-\lceil x \rceil|)</math> is the distance to the nearest integer.
Formulation and explanation
This means the following: take a point (α, β) in the plane, and then consider the sequence of points
- (2α, 2β), (3α, 3β), ... .
For each of these, multiply the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values will converge; it typically does not, in fact. The conjecture states something about the limit inferior, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.
- o(1/n)
in the little-o notation.
Connection to further conjectures
In 1955 Cassels and Swinnerton-Dyer.<ref>Template:Cite journal</ref> showed that Littlewood's Conjecture would follow from the following conjecture in the geometry of numbers in the case <math>n=3</math>:
Conjecture 1: Let L be the product of n linear forms on <math>\mathbb{R}^n</math>. Suppose <math>n\geq 3</math> and L is not a multiple of a form with integer coefficients. Then <math>\inf\{|L(x)|\mid x\in\mathbb{Z}^n\setminus\{0\}\}=0</math>.
Conjecture 1 is equivalent to the following conjecture concerning the orbits of the diagonal subgroup D on <math>SL(n, \mathbb{R})/SL(n, \mathbb{Z}),</math> as was essentially noticed by Cassels and Swinnerton-Dyer.
Conjecture 2: Let <math>n\geq 3</math>. For any <math>x\in SL(n, \mathbb{R})/SL(n, \mathbb{Z})</math>, if the orbit <math>Dx </math> is relatively compact, then <math>Dx</math> is closed.
This is due to Margulis. <ref>Template:Cite book</ref> 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, <math>\Gamma</math> a lattice in G, and H a closed connected subgroup generated by <math>(Ad_G, \mathbb{R})</math>-split elements, i.e. all eigenvalues of <math>Ad_G(g)</math> are real for each generator g. Then for any <math>x\in G/\Gamma</math>, exactly one of the following holds:
- <math>\overline{Hx}</math> is homogeneous, i.e. there is a closed subgroup F of G such that <math>\overline{Hx}=Fx</math>.
- There exists a closed connected subgroup F of G and a continuous epimorphism <math>\phi</math> from F onto a Lie group L such that <math>H\subset F</math>, <math>Fx</math> is closed in <math>G/\Gamma</math>, <math>\phi(F_x)</math> is closed in L where <math>F_x</math> is the stabilizer, and <math>\phi(H)</math> is a one-parameter subgroup of L containing no non-trivial <math>Ad_L</math>-unipotent elements, i.e. elements g for which 1 is the only eigenvalue of <math>Ad_L(g)</math>.
Partial results
Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero.<ref name=AB444>Adamczewski & Bugeaud (2010) p.444</ref> Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown<ref>Template:Cite journal</ref> that it must have Hausdorff dimension zero;<ref name=AB445>Adamczewski & Bugeaud (2010) p.445</ref> and in fact is a union of countably many compact sets of box-counting dimension zero. The result was proved by using a measure classification theorem for diagonalizable actions of higher-rank groups, and an isolation theorem proved by Lindenstrauss and Barak Weiss.
These results imply that non-trivial pairs (i.e., pairs (α,β) which are individually badly approximable and where 1, α, and β are linearly independent over <math>\mathbb{Q}</math>) satisfying the conjecture exist: indeed, given a real number α such that <math>\inf_{n \ge 1} n \cdot || n \alpha || > 0 </math>, it is possible to construct an explicit β such that (α,β) is non-trivial and satisfies the conjecture.<ref name=AB446>Adamczewski & Bugeaud (2010) p.446</ref>