Vol. 91, No. 1, 1980

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The best two-dimensional Diophantine approximation constant for cubic irrationals

William Wells Adams

Vol. 91 (1980), No. 1, 29–30
Abstract

Let 1, β1, β2 be a basis of a real cubic number field K. Let c0 = c0(β1β2) be the infimum over all constants c > 0 such that

|qβ1 − p1| < (c∕q)1∕2,  |qβ2 − p2| < (c∕q)1∕2

has an infinite number of solutions in integers q > 0, p1, p2. Set

C0 = sup c0(β1,β2).
β1,β2

The purpose of this note is to observe that combining a recent beautiful result in the geometry of numbers of A. C. Woods with the earlier work of the author, we obtain

Theorem. CO = 27.

Mathematical Subject Classification
Primary: 10E15, 10E15
Secondary: 10F10
Milestones
Received: 25 June 1980
Published: 1 November 1980
Authors
William Wells Adams