Oppenheim conjecture
Encyclopedia
In Diophantine approximation
, the Oppenheim conjecture concerns representations of numbers by real quadratic form
s in several variables. It was formulated in 1929 by Alexander Oppenheim
and later the conjectured property was further strengthened by Davenport and Oppenheim. Initial research on this problem took the number n of variables to be large, and applied a version of the Hardy-Littlewood circle method
. The definitive work of Margulis
, settling the conjecture in the affirmative, used methods arising from ergodic theory
and the study of discrete subgroups of semisimple Lie groups.
states that an indefinite integral quadratic form Q in n variables, n ≥ 5, nontrivially represents zero, i.e. there exists an non-zero vector x with integer components such that Q(x) = 0. The Oppenheim conjecture can be viewed as an analogue of this statement for forms Q that are not multiples of a rational form. It states that in this case, the set of values of Q on integer vectors is a dense subset of the real line
.
.
For n ≥ 5 this was conjectured by Oppenheim in 1929; the stronger version is due to Davenport in 1946.
This was conjectured by Oppenheim in 1953 and proved by Birch, Davenport, and Ridout for n at least 21, and by Davenport and Heilbronn for diagonal forms in five variables. Other partial results are due to Oppenheim (for forms in four variables,
but under the strong restriction that the form represents zero over Z), Watson, Iwaniec, Baker–Schlickewey. Early work analytic number theory
and reduction theory of quadratic forms.
The conjecture was proved in 1987 by Margulis in complete generality using methods of ergodic theory. Geometry of actions of certain unipotent subgroups of the orthogonal group
on the homogeneous space
of the lattices
in R3plays decisive role in this approach. It is sufficient to established the case n = 3. The idea to derive the Oppenheim conjecture from a statement about homogeneous group actions is usually attributed to M. S. Raghunathan, who observed in the 1970s that the conjecture for n = 3 is equivalent to the following property of the space of lattices:
However, Margulis later remarked that in an implicit form of this equivalence occurred already in a 1955 paper of Cassels
and H. P. F. Swinnerton-Dyer, albeit in a different language.
Shortly after Margulis's breakthrough, the proof was simplified and generalized by Dani and Margulis. Qualitative versions of the Oppenheim conjecture were later proved by Eskin–Margulis–Mozes. Borel
and Prasad
established some S-arithmetic analogues. The study of the properties of unipotent and quasiunipotent flows on homogeneous spaces remains an active area of research, with applications to further questions in the theory of 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....
, the Oppenheim conjecture concerns representations of numbers by real quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s in several variables. It was formulated in 1929 by Alexander Oppenheim
Alexander Oppenheim
Sir Alexander Oppenheim, OBE FRSE Knight Bachelor PMN was a British mathematician and philanthropist. In mathematics, his most notable contribution is his Oppenheim conjecture...
and later the conjectured property was further strengthened by Davenport and Oppenheim. Initial research on this problem took the number n of variables to be large, and applied a version of the Hardy-Littlewood circle method
Hardy-Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E...
. The definitive work 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...
, settling the conjecture in the affirmative, used methods arising from ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
and the study of discrete subgroups of semisimple Lie groups.
Short description
Meyer's theoremMeyer's theorem
In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation...
states that an indefinite integral quadratic form Q in n variables, n ≥ 5, nontrivially represents zero, i.e. there exists an non-zero vector x with integer components such that Q(x) = 0. The Oppenheim conjecture can be viewed as an analogue of this statement for forms Q that are not multiples of a rational form. It states that in this case, the set of values of Q on integer vectors is a dense subset of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
.
History
Several versions of the conjecture were formulated by Oppenheim and DavenportHarold Davenport
Harold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...
.
- Let Q be a real nondegenerate indefinite quadratic form in n variables. Suppose that n ≥ 3 and Q is not a multiple of a form with rational coefficients. Then for any ε > 0 there exists a non-zero vector x with integer components such that |Q(x)| < ε.
For n ≥ 5 this was conjectured by Oppenheim in 1929; the stronger version is due to Davenport in 1946.
- Let Q and n have the same meaning as before. Then for any ε > 0 there exists a non-zero vector x with integer components such that 0 < |Q(x,x)| < ε.
This was conjectured by Oppenheim in 1953 and proved by Birch, Davenport, and Ridout for n at least 21, and by Davenport and Heilbronn for diagonal forms in five variables. Other partial results are due to Oppenheim (for forms in four variables,
but under the strong restriction that the form represents zero over Z), Watson, Iwaniec, Baker–Schlickewey. Early work analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
and reduction theory of quadratic forms.
The conjecture was proved in 1987 by Margulis in complete generality using methods of ergodic theory. Geometry of actions of certain unipotent subgroups of the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
on the homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
of the lattices
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...
in R3plays decisive role in this approach. It is sufficient to established the case n = 3. The idea to derive the Oppenheim conjecture from a statement about homogeneous group actions is usually attributed to M. S. Raghunathan, who observed in the 1970s that the conjecture for n = 3 is equivalent to the following property of the space of lattices:
- Any relatively compact orbit of SO(2,1) in SL(3,R)/SL(3,Z) is compact.
However, Margulis later remarked that in an implicit form of this equivalence occurred already in a 1955 paper of 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 H. P. F. Swinnerton-Dyer, albeit in a different language.
Shortly after Margulis's breakthrough, the proof was simplified and generalized by Dani and Margulis. Qualitative versions of the Oppenheim conjecture were later proved by Eskin–Margulis–Mozes. Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
and Prasad
Gopal Prasad
Gopal Prasad is an Indian mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups.He is the Raoul Bott Professor of Mathematics at...
established some S-arithmetic analogues. The study of the properties of unipotent and quasiunipotent flows on homogeneous spaces remains an active area of research, with applications to further questions in the theory of 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....
.