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

Jeffrey C. Lagarias

Vol. 102 (1982), No. 1, 61–88

DOI: 10.2140/pjm.1982.102.61

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