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The Markoff and Lagrange Spectra
Thomas W. Cusick and Mary E. Flahive
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Mathematical Surveys and Monographs
1989; 97 pp; hardcover
Volume: 30
ISBN-10: 0-8218-1531-8
ISBN-13: 978-0-8218-1531-1
List Price: US$60
Member Price: US$48
Order Code: SURV/30
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This book is directed at mathematicians interested in Diophantine approximation and the theory of quadratic forms and the relationship of these subjects to Markoff and Lagrange spectra. The authors have gathered and systemized numerous results from the diverse and scattered literature, much of which has appeared in rather inaccessible Russian publications.

Readers will find a comprehensive overview of the theory of the Markoff and Lagrange spectra, starting with the origins of the subject in two papers of A. Markoff from 1879-80. Most of the progress since that time has occurred in the last 20 years or so, when there has been a resurgence of interest in these spectra. The authors provide an excellent exposition of these developments, in addition to presenting many proofs and correcting various errors in the literature.

Table of Contents

  • Older results on the Markoff spectrum
  • Markoff numbers and Markoff forms
  • The Markoff and Lagrange spectra compared
  • Hall's ray
  • Gaps in the spectra
  • The measure of the spectra
  • The spectra via the modular group
  • Alternative definitions of the spectra
  • Facts about continuants
  • Pell equations and automorphs of indefinite quadratic forms
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