Littlewood conjecture

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 $\alpha$ and $\beta$,

$\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}n\Vert n\alpha \Vert \Vert n\beta \Vert =0,$

where $\Vert x\Vert =\min (|x-\lfloor x\rfloor |,|x-\lceil x\rceil |)$ 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. 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 ${ℝ}^n$. Suppose $n\ge 3$ 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 $SL(n,ℝ)/SL(n,\mathbb{Z}),$ as was essentially noticed by Cassels and Swinnerton-Dyer.

Conjecture 2: Let $n\ge 3$. For any $x\in SL(n,ℝ)/SL(n,\mathbb{Z})$, 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 $(A{d}_G,ℝ)$-split elements, i.e. all eigenvalues of $A{d}_G(g)$ are real for each generator g. Then for any $x\in G/\Gamma$, exactly one of the following holds:

1. $\overline{Hx}$ is homogeneous, i.e. there is a closed subgroup F of G such that $\overline{Hx}=Fx$.

1. There exists a closed connected subgroup F of G and a continuous epimorphism $\varphi$ from F onto a Lie group L such that $H\subset F$, $Fx$ is closed in $G/\Gamma$, $\varphi (F_x)$ is closed in L where $F_x$ is the stabilizer, and $\varphi (H)$ is a one-parameter subgroup of L containing no non-trivial $A{d}_L$-unipotent elements, i.e. elements g for which 1 is the only eigenvalue of $A{d}_L(g)$.

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. Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown that it must have Hausdorff dimension zero; 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 $\mathbb{Q}$) satisfying the conjecture exist: indeed, given a real number α such that $\underset{n\ge 1}{\inf }n\cdot ||n\alpha ||>0$, it is possible to construct an explicit β such that (α,β) is non-trivial and satisfies the conjecture.

See also

• Littlewood polynomial