The two-dimensional Diophantine approximation constant. II

Thomas W. Cusick

Vol. 105 (1983), No. 1, 53–67

DOI: 10.2140/pjm.1983.105.53

Abstract

Given real numbers α and β, let c₁(α,β) denote the Diophantine approximation constant for the linear form x + αy + βz and let c₂(α,β) denote the corresponding dual constant for the simultaneous approximation of α and β. The paper gives various results about these constants in the case where α and β lie in some real cubic field. For example, it is shown that the suprema of c₁(α,β) and c₂(α,β), taken over all α, β such that 1, α, β is an integral basis for a real cubic field, are equal, and a necessary and sufficient condition for this common value to be equal to 2/7 is given.

Mathematical Subject Classification

Primary: 10F10, 10F10

Milestones

Received: 30 July 1981

Published: 1 March 1983