# Littlewood conjecture

In mathematics, the **Littlewood conjecture** is an open problem (as of 2016^{[update]}) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,

- [math]\displaystyle{ \liminf_{n\to\infty} \ n\,\Vert n\alpha\Vert \,\Vert n\beta\Vert = 0, }[/math]

where [math]\displaystyle{ \Vert \,\Vert }[/math] is here 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

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.

## 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.^{[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.

These results imply that non-trivial pairs satisfying the conjecture exist: indeed, given a real number α such that [math]\displaystyle{ \inf_{n \ge 1} n \cdot || n \alpha || \gt 0 }[/math], it is possible to construct an explicit β such that (α,β) satisfies the conjecture.^{[5]}

## See also

## References

- ↑ J.W.S. Cassels; H.P.F. Swinnerton-Dyer (1955-06-23). "On the product of three homogeneous linear forms and the indefinite ternary quadratic forms".
*Philosophical Transactions of the Royal Society A***248**(940): 73–96. doi:10.1098/rsta.1955.0010. Bibcode: 1955RSPTA.248...73C. - ↑ Adamczewski & Bugeaud (2010) p.444
- ↑ M. Einsiedler; A. Katok; E. Lindenstrauss (2006-09-01). "Invariant measures and the set of exceptions to Littlewood's conjecture".
*Annals of Mathematics***164**(2): 513–560. doi:10.4007/annals.2006.164.513. - ↑ Adamczewski & Bugeaud (2010) p.445
- ↑ Adamczewski & Bugeaud (2010) p.446

- Adamczewski, Boris; Bugeaud, Yann (2010). "8. Transcendence and diophantine approximation". in Berthé, Valérie; Rigo, Michael.
*Combinatorics, automata, and number theory*. Encyclopedia of Mathematics and its Applications.**135**. Cambridge:*Cambridge University Press*. pp. 410–451. ISBN 978-0-521-51597-9.

## Further reading

- Akshay Venkatesh (2007-10-29). "The work of Einsiedler, Katok, and Lindenstrauss on the Littlewood conjecture".
*Bull. Amer. Math. Soc. (N.S.)***45**(1): 117–134. doi:10.1090/S0273-0979-07-01194-9. http://www.cims.nyu.edu/~venkatesh/research/eklexp.pdf. Retrieved 2011-03-27.

Original source: https://en.wikipedia.org/wiki/ Littlewood conjecture.
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