Vol. 102, No. 1, 1982

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Best simultaneous Diophantine approximations. II. Behavior of consecutive best approximations

Jeffrey C. Lagarias

Vol. 102 (1982), No. 1, 61–88
Abstract

It is well-known that the best Diophantine approximations to a single real number 𝜃 are exactly the convergents of the continued fraction expansion of 𝜃. The properties of one-dimensional best approximations that make this true are shown not to hold in general for best simultaneous Diophantine approximations to α Rn when n 2. They do hold in a weak form for all badly approximate vectors α Rn.

Mathematical Subject Classification
Primary: 10F10, 10F10
Secondary: 10F20
Milestones
Received: 15 July 1980
Published: 1 September 1982
Authors
Jeffrey C. Lagarias
University of Michigan
United States