Littlewood conjecture
Encyclopedia
In mathematics
, the Littlewood conjecture is an open problem
in Diophantine approximation
, posed by John Edensor Littlewood
around 1930. It states that for any two real number
s α and β,
where
is here the distance to the nearest integer.
, (3α,3β), ... .
For each of these consider the closest 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 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.
in the little-o notation.
, 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. 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 = SLn(R), Γ = SLn(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 Margulis
on Lie group
s.
have shown that it must have Hausdorff dimension
zero; and in fact is a union of countably many compact set
s of 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Littlewood conjecture is an open problem
Open problem
In science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved . Some questions remain unanswered for centuries before solutions are found...
in Diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
, posed by John Edensor Littlewood
John Edensor Littlewood
John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
around 1930. It states that for any two real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s α and β,
where
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, (3α,3β), ... .
For each of these consider the closest lattice point
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
, 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 converge; it typically does not, in fact. The conjecture states something about the limit inferior
Limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence...
, 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 numbersGeometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
, about the minimum on a non-zero lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
point of a product of three linear forms in three real variables: the implication was shown in 1955 by J. W. S. Cassels
J. W. S. Cassels
John William Scott Cassels , FRS is a leading English mathematician.-Biography:Educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh, Cassels graduated from the University of Edinburgh with an MA in 1943.His academic career was interrupted in World War II...
and Swinnerton-Dyer. 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 = SLn(R), Γ = SLn(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 Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
on Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s.
Partial results
Progress has been made in showing that the exceptional set of real pairs (α,β) violating the statement of the conjecture must be small. Manfred Einsiedler, Anatole Katok and Elon LindenstraussElon Lindenstrauss
Elon Lindenstrauss is an Israeli mathematician, and a winner of the 2010 Fields Medal.Since 2004, he has been a professor at Princeton University...
have shown that it must have Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
zero; and in fact is a union of countably many compact set
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
s of 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.