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In [[mathematics]], the '''Littlewood conjecture''' is an [[open problem]] ({{As of|2011|lc=on}}) in [[Diophantine approximation]], proposed by [[John Edensor Littlewood]] around 1930. It states that for any two [[real number]]s α and β,
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:<math>\liminf_{n\to\infty} \ n\,\Vert n\alpha\Vert \,\Vert n\beta\Vert = 0,</math>
 
where <math>\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 consider the closest [[Lattice (group)|lattice point]], as determined by multiplying 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 [[Convergence (mathematics)|converge]]; it typically does not, in fact. The conjecture states something about the [[Limit superior and limit inferior|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 [[Big_O_notation#Little-o_notation|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 (group)|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]].<ref>{{cite journal | author=J.W.S. Cassels, H.P.F. Swinnerton-Dyer | title=On the product of three homogeneous linear forms and the indefinite ternary quadratic forms | journal=[[Philosophical Transactions of the Royal Society A]] | volume=248 | issue=940 | date=1955-06-23 | pages=73–96 | doi=10.1098/rsta.1955.0010 | jstor=91633 | mr = 70653 |zbl=0065.27905}}</ref> 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<sub>n</sub>''(''R''), Γ = ''SL<sub>n</sub>''(''Z''), and the subgroup ''D'' of ''G'' of [[diagonal matrices]].
 
'''''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 [[Grigory Margulis|Margulis]] on [[Lie group]]s.
 
==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>{{cite journal | author=M. Einsiedler, A. Katok, E. Lindenstrauss | title=Invariant measures and the set of exceptions to Littlewood's conjecture | journal=[[Annals of Mathematics]] | volume=164 | issue=2 | pages=513–560 | date=2006-09-01 | doi=10.4007/annals.2006.164.513 | arxiv=math.DS/0612721 | mr = 2247967 |zbl=1109.22004}}</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 space|compact set]]s of [[Minkowski–Bouligand dimension|box-counting dimension]] zero. The result was proved by using 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>\inf_{n \ge 1} n \cdot || n \alpha || > 0 </math>, it is possible to construct an explicit β such that (α,β) satisfies the conjecture.<ref name=AB446>Adamczewski & Bugeaud (2010) p.446</ref>
 
==See also==
* [[Littlewood polynomial]]
 
==References==
{{reflist}}
*{{cite book | last1=Adamczewski | first1=Boris | last2=Bugeaud | first2=Yann | chapter=8. Transcendence and diophantine approximation | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michael | title=Combinatorics, automata, and number theory | location=Cambridge | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | volume=135 | pages=410–451 | year=2010 | isbn=978-0-521-51597-9 | zbl=pre05879512 }}
 
==Further reading==
* {{cite journal | author=Akshay Venkatesh | title=The work of Einsiedler, Katok, and Lindenstrauss on the Littlewood conjecture | journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc. (N.S.)]] | volume=45 | issue=1 | date=2007-10-29 | pages=117–134 | url=http://www.cims.nyu.edu/~venkatesh/research/eklexp.pdf | doi=10.1090/S0273-0979-07-01194-9 | accessdate=2011-03-27 | mr=2358379 | zbl = 1194.11075 }}
 
{{DEFAULTSORT:Littlewood Conjecture}}
[[Category:Diophantine approximation]]
[[Category:Conjectures]]

Latest revision as of 01:35, 13 December 2014

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