In mathematics, the subspace theorem says that points of small

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

lie in a finite number of

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s. It is a result obtained by .

Statement

The subspace theorem states that if ''L''₁,...,''L''_''n'' are

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...

forms Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form also refers to: *Form (document), a document (printed or electronic) with spaces in which to write or enter data * ...

in ''n'' variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points ''x'' with :

, L_1(x)\cdots L_n(x), <, x, ^

lie in a finite number of proper subspaces of Q^''n''. A quantitative form of the theorem, in which the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by to allow more general absolute values on

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

Applications

The theorem may be used to obtain results on

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s such as

Siegel's theorem on integral points In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve ''C'' of genus ''g'' defined over a number field ''K'', presented in affine space in a given coordinate system, there are only finitely many points on ''C ...

and solution of the

S-unit equation In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units. Definition Let ''K'' be a number ...

.Bombieri & Gubler (2006) pp. 176–230.

A corollary on Diophantine approximation

The following corollary to the subspace theorem is often itself referred to as the ''subspace theorem''. If ''a''₁,...,''a''_''n'' are algebraic such that 1,''a''₁,...,''a''_''n'' are linearly independent over Q and ε>0 is any given real number, then there are only finitely many rational ''n''-tuples (''x''₁/y,...,''x''_''n''/y) with :

, a_i-x_i/y, The specialization ''n'' = 1 gives the

Thue–Siegel–Roth theorem In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a ...

. One may also note that the exponent 1+1/''n''+ε is best possible by

Dirichlet's theorem on diophantine approximation In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and ...

References

* * * * * {{cite book , last=Schmidt , first=Wolfgang M. , authorlink=Wolfgang M. Schmidt , title=Diophantine approximations and Diophantine equations , series=Lecture Notes in Mathematics , volume=1467 , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, year=1991 , location=Berlin , isbn=3-540-54058-X , zbl=0754.11020 , mr=1176315 , doi=10.1007/BFb0098246, s2cid=118143570 Diophantine approximation Theorems in number theory