An introduction to diophantine approximation / J. W. S. CasselsType de document : MonographieCollection : Cambridge tracts in mathematics and mathematical physics, 45Langue : anglais.Pays : Grande Bretagne.Éditeur : Cambridge : Cambridge University Press, 1965Description : 1 vol. (X-168 p.) : ill. ; 22 cmISSN : 0068-6824.Bibliographie : Bibliogr. p. 163-168. Index.Sujet MSC : 11Jxx, Number theory, Diophantine approximation, transcendental number theory
11Hxx, Number theory, Geometry of numbers
11Kxx, Number theory, Probabilistic theory: distribution modulo 1; metric theory of algorithmsEn-ligne : Google edition | MathSciNet
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Chapter headings: I. Homogeneous approximation; II. The Markoff chain; III. Inhomogeneous approximation; IV. Uniform distribution; V. Transference theorems; VI. Rational approximation to algebraic numbers; VII. Metrical theory; VIII. The Pisot-Vijayaraghavan numbers; and Appendices.
This tract is rather more than an introduction to the subject of Diophantine approximation, and is best described as a concise textbook. It contains a remarkable wealth of material, and most of the major theories of the subject are treated fairly fully. The author has taken much trouble to present the proofs in small compass, though in consequence it is not always possible to follow the general principle of a proof without mastering the details.
Chapter II contains all the main results of Markoff's theory of the minima of indefinite binary quadratic forms, proved in a mere 25 pages. Chapter III gives a full account of what is known in connection with Kronecker's theorem. Chapter IV contains the substance of Weyl's great memoir of 1916, in a modern presentation. Chapter V is based on the work of Khintchine, with recent developments by Hlawka, the author and Birch. Chapter VI contains the proof of K. F. Roth's remarkable recent theorem. Chapters VII and VIII are well-integrated accounts of work which was previously scattered in the literature.
Everyone interested in the subject has reason to be grateful to the author for this valuable work. It should be studied in conjunction with Koksma's report (Diophantische Approximationen, Springer, Berlin, 1936), which surveys a somewhat wider field, but for the most part without proofs. (MathSciNet)
Bibliogr. p. 163-168. Index