# Littlewood conjecture

In mathematics, the **Littlewood conjecture** is an open problem (as of May 2021) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,

This means the following: take a point (α,β) in the plane, and then consider the sequence of points

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.

It is known that this would follow from a result in the geometry of numbers, about the minimum on a non-zero lattice point of a product of three linear forms in three real variables: the implication was shown in 1955 by J. W. S. Cassels and Swinnerton-Dyer.^{[1]} This can be formulated another way, in group-theoretic terms. There is now another conjecture, expected to hold for *n* ≥ 3: it is stated in terms of *G* = *SL _{n}*(

*R*), Γ =

*SL*(

_{n}*Z*), and the subgroup

*D*of diagonal matrices in

*G*.

* Conjecture*: for any

*g*in

*G*/Γ such that

*Dg*is relatively compact (in

*G*/Γ), then

*Dg*is closed.

This in turn is a special case of a general conjecture of Margulis on Lie groups.

Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero.^{[2]} Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown^{[3]} that it must have Hausdorff dimension zero;^{[4]} 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.