Transference theorem in Diophantine approximation
An assertion on a relation between the solvability in integers of a system of inequalities and the solvability of another system related in a definite way to the first. A classical example of a linear transference theorem is Khinchin's transference principle (see Diophantine approximations). More general linear transference theorems concern the relations between solutions in integers of a system of homogeneous linear inequalities having a non-singular square matrix and solutions of the corresponding system with the inverse transposed matrix: The existence of a non-trivial solution of one system guarantees the existence of such for the other, and vice versa. Such links exist between linear homogeneous and inhomogeneous systems of inequalities, where the absence of non-trivial solutions of the homogeneous system guarantees the existence of solutions of the corresponding inhomogeneous systems, and vice versa. Similar relations are known for non-linear problems, but they are less definitely expressed and have been rarely examined. The major importance of transference theorems in the theory of Diophantine approximation can be explained by a transference theorem in the geometry of numbers: For convex sets, there are relationships between the presence of integer points in the given set and in the set reciprocal to it.
|||J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)|
|||J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1959)|
Transference theorem in Diophantine approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transference_theorem_in_Diophantine_approximation&oldid=34033