Vol. 105, No. 1, 1983

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The two-dimensional Diophantine approximation constant. II

Thomas W. Cusick

Vol. 105 (1983), No. 1, 53–67

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
Received: 30 July 1981
Published: 1 March 1983
Thomas W. Cusick
Department of Mathematics
SUNY Buffalo
College of Arts and Sciences
Buffalo NY 14260-2900
United States