Diophantine Approximations

By: IVAN NIVEN

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Paperback | 14 March 2008

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78 Pages

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This self-contained treatment originated as a series of lectures delivered to the Mathematical Association of America. It covers basic results on homogeneous approximation of real numbers; the analogue for complex numbers; basic results for nonhomogeneous approximation in the real case; the analogue for complex numbers; and fundamental properties of the multiples of an irrational number, for both the fractional and integral parts.
The author refrains from the use of continuous fractions and includes basic results in the complex case, a feature often neglected in favor of the real number discussion. Each chapter concludes with a bibliographic account of closely related work; these sections also contain the sources from which the proofs are drawn.

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Non-Fiction Mathematics Number Theory Groups & Group Theory

Notation and Conventions	p. viii
The Approximation of Irrationals by Rationals
The pigeon-hole principle	p. 1
The theorem of Hurwitz	p. 3
Asymmetric approximation	p. 9
Further results	p. 13
The Product of Linear Forms
The Minkowski results	p. 16
Further results	p. 21
The Multiples of an Irrational Number
A sequence of rational approximations to an irrational number	p. 23
The uniform distribution of the fractional parts	p. 24
The uniform distribution of the integral parts	p. 27
Kronecker's theorem	p. 28
Results of Skolem and Bang	p. 34
Sets defined by rational numbers	p. 38
Further results	p. 44
The Approximation of Complex Numbers
Ford's theorem	p. 46
Further results	p. 53
The Product of Complex Linear Forms
The covering of lattice points	p. 54
An inequality in the complex plane	p. 56
Minimum values of forms	p. 60
Further results	p. 61
Bibliography	p. 63
Index	p. 67
Table of Contents provided by Ingram. All Rights Reserved.

Diophantine Approximations

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