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Khintchine’s singular Diophantine systems and their applications. (English. Russian original) Zbl 1225.11094
Russ. Math. Surv. 65, No. 3, 433-511 (2010); translation from Uspekhi Mat. Nauk. 65, No. 3, 43-126 (2010).
In this comprehensive review the author describes Khintchine’s methods related to the existence of vectors of real numbers that are well approximable by rationals. Starting from so-called singular systems introduced by A. Khintchine in 1926 [Math. Z. 24, 706–714 (1926; JFM 52.0183.02)], he ends up with some very recent results on lacunary sequences related to the Erdős conjecture stated in 1953 (which was in fact solved by Khintchine in 1926). An important part in this review is dedicated to the problems of simultaneous homogeneous and inhomogeneous approximation, transference principle and some problems of metric number theory. Applications of these methods (in lattices, dynamical systems, Fourier analysis, Schmidt’s \((\alpha,\beta)\)-games, etc.) are also described. Some of the 74 theorems are given with proofs. In particular, the author gives a version of Jarník’s theorem on Chebyshev systems with explicit constants (see Theorem 29). The paper contains 145 references.

11J13 Simultaneous homogeneous approximation, linear forms
11J83 Metric theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
JFM 52.0183.02
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