He co-edited " Analytic Number Theory ", a tome about prime numbers, divisor problems, Diophantine equations, and other topics related to analytic number theory, including Diophantine approximations, and the theory of zeta and L-functions.
32.
This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations : in fact an immediate consequence is that for a given irrational ?, the inequality
33.
They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results " ad hoc " methods, in some sense inspired by the Pad?theory, typically replace them.
34.
Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting ? = 0 : by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case.
35.
Khinchin made significant contributions to the metric theory of Diophantine approximations and established an important result for simple real continued fractions, discovering a property of such numbers that leads to what is now known as Khinchin's constant.
36.
This was proved by combining a version of the Thue Siegel Roth theorem, from diophantine approximation, with the Mordell Weil theorem from diophantine geometry ( required in Weil's version, to apply to the Jacobian variety of " C " ).
37.
He wrote a Ph . D . in diophantine approximation under J . E . Littlewood and G . H . Hardy at the University of Cambridge, completed in 1939 . He had positions at MIT and Stanford before his appointment in 1950 at Princeton University.
38.
Informally, for every point in " X ", the point is either in " A " or arbitrarily " close " to a member of " A " & mdash; for instance, every real number is either a rational number or has one arbitrarily close to it ( see Diophantine approximation ).
39.
An example would be a planetary system, with planets in orbits moving with theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy : if we wait long enough we can observe the planets all return to within a second of arc to the positions they once were in.
40.
He extended the theory developed by Paul Vojta ( an analogy of the Nevanlinna theory, part of the value distribution theory of holomorphic functions, to diophantine geometry ) and applied the method of " dynamic diophantine approximation " which he developed in the process, to transcendental algebraic geometry ( and therefore for varieties over the complex numbers, where methods of complex analysis can be used ).
